author | Kevin Walker <kevin@canyon23.net> |
Mon, 30 Aug 2010 08:54:01 -0700 | |
changeset 526 | 803cc581fd42 |
parent 524 | edf8798ef477 |
child 527 | 19e58f33cdc3 |
permissions | -rw-r--r-- |
70 | 1 |
%!TEX root = ../blob1.tex |
2 |
||
520 | 3 |
\section{Action of \texorpdfstring{$\CH{X}$}{C*(Homeo(M))}} |
100
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
92
diff
changeset
|
4 |
\label{sec:evaluation} |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
92
diff
changeset
|
5 |
|
513 | 6 |
In this section we extend the action of homeomorphisms on $\bc_*(X)$ |
7 |
to an action of {\it families} of homeomorphisms. |
|
8 |
That is, for each pair of homeomorphic manifolds $X$ and $Y$ |
|
9 |
we define a chain map |
|
10 |
\[ |
|
11 |
e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) , |
|
12 |
\] |
|
13 |
where $CH_*(X, Y) = C_*(\Homeo(X, Y))$, the singular chains on the space |
|
14 |
of homeomorphisms from $X$ to $Y$. |
|
15 |
(If $X$ and $Y$ have non-empty boundary, these families of homeomorphisms |
|
526 | 16 |
are required to restrict to a fixed homeomorphism on the boundaries.) |
17 |
These actions (for various $X$ and $Y$) are compatible with gluing. |
|
513 | 18 |
See \S \ref{ss:emap-def} for a more precise statement. |
19 |
||
20 |
The most convenient way to prove that maps $e_{XY}$ with the desired properties exist is to |
|
526 | 21 |
introduce a homotopy equivalent alternate version of the blob complex, $\btc_*(X)$, |
513 | 22 |
which is more amenable to this sort of action. |
23 |
Recall from Remark \ref{blobsset-remark} that blob diagrams |
|
24 |
have the structure of a sort-of-simplicial set. |
|
25 |
Blob diagrams can also be equipped with a natural topology, which converts this |
|
26 |
sort-of-simplicial set into a sort-of-simplicial space. |
|
27 |
Taking singular chains of this space we get $\btc_*(X)$. |
|
28 |
The details are in \S \ref{ss:alt-def}. |
|
526 | 29 |
We also prove a useful lemma (\ref{small-blobs-b}) which says that we can assume that |
30 |
blobs are small with respect to any fixed open cover. |
|
31 |
||
32 |
||
513 | 33 |
|
524
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
34 |
%Since $\bc_*(X)$ and $\btc_*(X)$ are homotopy equivalent one could try to construct |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
35 |
%the $CH_*$ actions directly in terms of $\bc_*(X)$. |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
36 |
%This was our original approach, but working out the details created a nearly unreadable mess. |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
37 |
%We have salvaged a sketch of that approach in \S \ref{ss:old-evmap-remnants}. |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
38 |
% |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
39 |
%\nn{should revisit above intro after this section is done} |
514
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
40 |
|
512
050dba5e7bdd
fixing some (but not all!?) of the hyperref warnings; start on revision of evmap
Kevin Walker <kevin@canyon23.net>
parents:
494
diff
changeset
|
41 |
|
050dba5e7bdd
fixing some (but not all!?) of the hyperref warnings; start on revision of evmap
Kevin Walker <kevin@canyon23.net>
parents:
494
diff
changeset
|
42 |
\subsection{Alternative definitions of the blob complex} |
513 | 43 |
\label{ss:alt-def} |
512
050dba5e7bdd
fixing some (but not all!?) of the hyperref warnings; start on revision of evmap
Kevin Walker <kevin@canyon23.net>
parents:
494
diff
changeset
|
44 |
|
514
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
45 |
\newcommand\sbc{\bc^{\cU}} |
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
46 |
|
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
47 |
In this subsection we define a subcomplex (small blobs) and supercomplex (families of blobs) |
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
48 |
of the blob complex, and show that they are both homotopy equivalent to $\bc_*(X)$. |
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
49 |
|
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
50 |
\medskip |
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
51 |
|
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
52 |
If $b$ is a blob diagram in $\bc_*(X)$, define the {\it support} of $b$, denoted |
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
53 |
$\supp(b)$ or $|b|$, to be the union of the blobs of $b$. |
526 | 54 |
%For a general $k$-chain $a\in \bc_k(X)$, define the support of $a$ to be the union |
55 |
%of the supports of the blob diagrams which appear in it. |
|
56 |
More generally, we say that a chain $a\in \bc_k(X)$ is supported on $S$ if |
|
57 |
$a = a'\bullet r$, where $a'\in \bc_k(S)$ and $r\in \bc_0(X\setmin S)$. |
|
514
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
58 |
|
526 | 59 |
Similarly, if $f: P\times X\to X$ is a family of homeomorphisms and $Y\sub X$, we say that $f$ is |
514
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
60 |
{\it supported on $Y$} if $f(p, x) = f(p', x)$ for all $x\in X\setmin Y$ and all $p,p'\in P$. |
526 | 61 |
%Equivalently, $f = f'\bullet r$, where $f'\in CH_k(Y)$ and $r\in CH_0(X\setmin Y)$. |
514
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
62 |
We will sometimes abuse language and talk about ``the" support of $f$, |
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
63 |
again denoted $\supp(f)$ or $|f|$, to mean some particular choice of $Y$ such that |
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
64 |
$f$ is supported on $Y$. |
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
65 |
|
515 | 66 |
If $f: M \cup (Y\times I) \to M$ is a collaring homeomorphism |
67 |
(cf. end of \S \ref{ss:syst-o-fields}), |
|
68 |
we say that $f$ is supported on $S\sub M$ if $f(x) = x$ for all $x\in M\setmin S$. |
|
69 |
||
526 | 70 |
\medskip |
71 |
||
514
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
72 |
Fix $\cU$, an open cover of $X$. |
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
73 |
Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(X)$ |
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
74 |
of all blob diagrams in which every blob is contained in some open set of $\cU$, |
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
75 |
and moreover each field labeling a region cut out by the blobs is splittable |
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
76 |
into fields on smaller regions, each of which is contained in some open set of $\cU$. |
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
77 |
|
524
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
78 |
\begin{lemma}[Small blobs] \label{small-blobs-b} \label{thm:small-blobs} |
514
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
79 |
The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. |
523 | 80 |
\end{lemma} |
514
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
81 |
|
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
82 |
\begin{proof} |
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
83 |
It suffices to show that for any finitely generated pair of subcomplexes |
524
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
84 |
\[ |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
85 |
(C_*, D_*) \sub (\bc_*(X), \sbc_*(X)) |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
86 |
\] |
514
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
87 |
we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$ |
524
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
88 |
and |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
89 |
\[ |
526 | 90 |
h\bd(x) + \bd h(x) - x \in \sbc_*(X) |
524
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
91 |
\] |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
92 |
for all $x\in C_*$. |
514
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
93 |
|
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
94 |
For simplicity we will assume that all fields are splittable into small pieces, so that |
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
95 |
$\sbc_0(X) = \bc_0$. |
515 | 96 |
(This is true for all of the examples presented in this paper.) |
514
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
97 |
Accordingly, we define $h_0 = 0$. |
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
98 |
|
515 | 99 |
Next we define $h_1$. |
514
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
100 |
Let $b\in C_1$ be a 1-blob diagram. |
515 | 101 |
Let $B$ be the blob of $b$. |
102 |
We will construct a 1-chain $s(b)\in \sbc_1$ such that $\bd(s(b)) = \bd b$ |
|
103 |
and the support of $s(b)$ is contained in $B$. |
|
104 |
(If $B$ is not embedded in $X$, then we implicitly work in some term of a decomposition |
|
105 |
of $X$ where $B$ is embedded. |
|
106 |
See \ref{defn:configuration} and preceding discussion.) |
|
107 |
It then follows from \ref{disj-union-contract} that we can choose |
|
108 |
$h_1(b) \in \bc_1(X)$ such that $\bd(h_1(b)) = s(b) - b$. |
|
109 |
||
110 |
Roughly speaking, $s(b)$ consists of a series of 1-blob diagrams implementing a series |
|
111 |
of small collar maps, plus a shrunken version of $b$. |
|
526 | 112 |
The composition of all the collar maps shrinks $B$ to a ball which is small with respect to $\cU$. |
515 | 113 |
|
519
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
114 |
Let $\cV_1$ be an auxiliary open cover of $X$, subordinate to $\cU$ and |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
115 |
also satisfying conditions specified below. |
515 | 116 |
Let $b = (B, u, r)$, $u = \sum a_i$ be the label of $B$, $a_i\in \bc_0(B)$. |
526 | 117 |
Choose a sequence of collar maps $\bar{f}_j:B\cup\text{collar}\to B$ satisfying conditions which we cannot express |
118 |
until introducing more notation. |
|
119 |
Let $f_j:B\to B$ be the restriction of $\bar{f}_j$ to $B$; $f_j$ maps $B$ homeomorphically to |
|
120 |
a slightly smaller submanifold of $B$. |
|
121 |
Let $g_j = f_1\circ f_2\circ\cdots\circ f_j$. |
|
122 |
Let $g$ be the last of the $g_j$'s. |
|
123 |
Choose the sequence $\bar{f}_j$ so that |
|
124 |
$g(B)$ is contained is an open set of $\cV_1$ and |
|
125 |
$g_{j-1}(|f_j|)$ is also contained is an open set of $\cV_1$. |
|
126 |
||
515 | 127 |
There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$ |
526 | 128 |
(more specifically, $|c_{ij}| = g_{j-1}(|f_j|)$) |
129 |
and $\bd c_{ij} = g_{j-1}(a_i) - g_{j}(a_i)$. |
|
515 | 130 |
Define |
131 |
\[ |
|
132 |
s(b) = \sum_{i,j} c_{ij} + g(b) |
|
133 |
\] |
|
134 |
and choose $h_1(b) \in \bc_1(X)$ such that |
|
135 |
\[ |
|
136 |
\bd(h_1(b)) = s(b) - b . |
|
137 |
\] |
|
138 |
||
139 |
Next we define $h_2$. |
|
516
ec5c68b275c0
making latest available to office computer
Kevin Walker <kevin@canyon23.net>
parents:
515
diff
changeset
|
140 |
Let $b\in C_2$ be a 2-blob diagram. |
ec5c68b275c0
making latest available to office computer
Kevin Walker <kevin@canyon23.net>
parents:
515
diff
changeset
|
141 |
Let $B = |b|$, either a ball or a union of two balls. |
ec5c68b275c0
making latest available to office computer
Kevin Walker <kevin@canyon23.net>
parents:
515
diff
changeset
|
142 |
By possibly working in a decomposition of $X$, we may assume that the ball(s) |
ec5c68b275c0
making latest available to office computer
Kevin Walker <kevin@canyon23.net>
parents:
515
diff
changeset
|
143 |
of $B$ are disjointly embedded. |
ec5c68b275c0
making latest available to office computer
Kevin Walker <kevin@canyon23.net>
parents:
515
diff
changeset
|
144 |
We will construct a 2-chain $s(b)\in \sbc_2$ such that |
ec5c68b275c0
making latest available to office computer
Kevin Walker <kevin@canyon23.net>
parents:
515
diff
changeset
|
145 |
\[ |
ec5c68b275c0
making latest available to office computer
Kevin Walker <kevin@canyon23.net>
parents:
515
diff
changeset
|
146 |
\bd(s(b)) = \bd(h_1(\bd b) + b) = s(\bd b) |
ec5c68b275c0
making latest available to office computer
Kevin Walker <kevin@canyon23.net>
parents:
515
diff
changeset
|
147 |
\] |
ec5c68b275c0
making latest available to office computer
Kevin Walker <kevin@canyon23.net>
parents:
515
diff
changeset
|
148 |
and the support of $s(b)$ is contained in $B$. |
ec5c68b275c0
making latest available to office computer
Kevin Walker <kevin@canyon23.net>
parents:
515
diff
changeset
|
149 |
It then follows from \ref{disj-union-contract} that we can choose |
ec5c68b275c0
making latest available to office computer
Kevin Walker <kevin@canyon23.net>
parents:
515
diff
changeset
|
150 |
$h_2(b) \in \bc_2(X)$ such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$. |
514
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
151 |
|
516
ec5c68b275c0
making latest available to office computer
Kevin Walker <kevin@canyon23.net>
parents:
515
diff
changeset
|
152 |
Similarly to the construction of $h_1$ above, |
ec5c68b275c0
making latest available to office computer
Kevin Walker <kevin@canyon23.net>
parents:
515
diff
changeset
|
153 |
$s(b)$ consists of a series of 2-blob diagrams implementing a series |
ec5c68b275c0
making latest available to office computer
Kevin Walker <kevin@canyon23.net>
parents:
515
diff
changeset
|
154 |
of small collar maps, plus a shrunken version of $b$. |
ec5c68b275c0
making latest available to office computer
Kevin Walker <kevin@canyon23.net>
parents:
515
diff
changeset
|
155 |
The composition of all the collar maps shrinks $B$ to a sufficiently small |
ec5c68b275c0
making latest available to office computer
Kevin Walker <kevin@canyon23.net>
parents:
515
diff
changeset
|
156 |
disjoint union of balls. |
ec5c68b275c0
making latest available to office computer
Kevin Walker <kevin@canyon23.net>
parents:
515
diff
changeset
|
157 |
|
519
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
158 |
Let $\cV_2$ be an auxiliary open cover of $X$, subordinate to $\cU$ and |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
159 |
also satisfying conditions specified below. |
517 | 160 |
As before, choose a sequence of collar maps $f_j$ |
161 |
such that each has support |
|
162 |
contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms |
|
163 |
yields an embedding $g:B\to B$ such that $g(B)$ is contained in an open set of $\cV_1$. |
|
164 |
Let $g_j:B\to B$ be the embedding at the $j$-th stage. |
|
519
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
165 |
|
517 | 166 |
Fix $j$. |
526 | 167 |
We will construct a 2-chain $d_j$ such that $\bd d_j = g_{j-1}(s(\bd b)) - g_{j}(s(\bd b))$. |
168 |
Let $s(\bd b) = \sum e_k$, and let $\{p_m\}$ be the 0-blob diagrams |
|
517 | 169 |
appearing in the boundaries of the $e_k$. |
170 |
As in the construction of $h_1$, we can choose 1-blob diagrams $q_m$ such that |
|
526 | 171 |
$\bd q_m = g_{j-1}(p_m) - g_j(p_m)$ and $\supp(q_m)$ is contained in an open set of $\cV_1$. |
519
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
172 |
If $x$ is a sum of $p_m$'s, we denote the corresponding sum of $q_m$'s by $q(x)$. |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
173 |
|
526 | 174 |
Now consider, for each $k$, $g_{j-1}(e_k) - q(\bd e_k)$. |
175 |
This is a 1-chain whose boundary is $g_j(\bd e_k)$. |
|
519
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
176 |
The support of $e_k$ is $g_{j-1}(V)$ for some $V\in \cV_1$, and |
520 | 177 |
the support of $q(\bd e_k)$ is contained in a union $V'$ of finitely many open sets |
178 |
of $\cV_1$, all of which contain the support of $f_j$. |
|
519
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
179 |
We now reveal the mysterious condition (mentioned above) which $\cV_1$ satisfies: |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
180 |
the union of $g_{j-1}(V)$ and $V'$, for all of the finitely many instances |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
181 |
arising in the construction of $h_2$, lies inside a disjoint union of balls $U$ |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
182 |
such that each individual ball lies in an open set of $\cV_2$. |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
183 |
(In this case there are either one or two balls in the disjoint union.) |
520 | 184 |
For any fixed open cover $\cV_2$ this condition can be satisfied by choosing $\cV_1$ |
185 |
to be a sufficiently fine cover. |
|
526 | 186 |
It follows from \ref{disj-union-contract} that we can choose |
187 |
$x_k \in \bc_2(X)$ with $\bd x_k = g_{j-1}(e_k) - g_j(e_k) - q(\bd e_k)$ |
|
519
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
188 |
and with $\supp(x_k) = U$. |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
189 |
We can now take $d_j \deq \sum x_k$. |
526 | 190 |
It is clear that $\bd d_j = \sum (g_{j-1}(e_k) - g_j(e_k)) = g_{j-1}(s(\bd b)) - g_{j}(s(\bd b))$, as desired. |
519
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
191 |
\nn{should maybe have figure} |
517 | 192 |
|
519
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
193 |
We now define |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
194 |
\[ |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
195 |
s(b) = \sum d_j + g(b), |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
196 |
\] |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
197 |
where $g$ is the composition of all the $f_j$'s. |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
198 |
It is easy to verify that $s(b) \in \sbc_2$, $\supp(s(b)) = \supp(b)$, and |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
199 |
$\bd(s(b)) = s(\bd b)$. |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
200 |
If follows that we can choose $h_2(b)\in \bc_2(X)$ such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$. |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
201 |
This completes the definition of $h_2$. |
517 | 202 |
|
519
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
203 |
The general case $h_l$ is similar. |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
204 |
When constructing the analogue of $x_k$ above, we will need to find a disjoint union of balls $U$ |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
205 |
which contains finitely many open sets from $\cV_{l-1}$ |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
206 |
such that each ball is contained in some open set of $\cV_l$. |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
207 |
For sufficiently fine $\cV_{l-1}$ this will be possible. |
520 | 208 |
Since $C_*$ is finite, the process terminates after finitely many, say $r$, steps. |
209 |
We take $\cV_r = \cU$. |
|
517 | 210 |
|
519
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
211 |
\nn{should probably be more specific at the end} |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
212 |
\end{proof} |
517 | 213 |
|
514
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
214 |
|
519
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
215 |
\medskip |
514
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
216 |
|
519
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
217 |
Next we define the sort-of-simplicial space version of the blob complex, $\btc_*(X)$. |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
218 |
First we must specify a topology on the set of $k$-blob diagrams, $\BD_k$. |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
219 |
We give $\BD_k$ the finest topology such that |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
220 |
\begin{itemize} |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
221 |
\item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous. |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
222 |
\item \nn{something about blob labels and vector space structure} |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
223 |
\item \nn{maybe also something about gluing} |
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
224 |
\end{itemize} |
514
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
225 |
|
519
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
226 |
Next we define $\btc_*(X)$ to be the total complex of the double complex (denoted $\btc_{**}$) |
521
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
227 |
whose $(i,j)$ entry is $C_j(\BD_i)$, the singular $j$-chains on the space of $i$-blob diagrams. |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
228 |
The vertical boundary of the double complex, |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
229 |
denoted $\bd_t$, is the singular boundary, and the horizontal boundary, denoted $\bd_b$, is |
519
1e56e60dcf15
first draft of new sm blobs; needs to be proof-read and revised
Kevin Walker <kevin@canyon23.net>
parents:
517
diff
changeset
|
230 |
the blob boundary. |
514
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
231 |
|
521
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
232 |
We will regard $\bc_*(X)$ as the subcomplex $\btc_{*0}(X) \sub \btc_{**}(X)$. |
520 | 233 |
The main result of this subsection is |
234 |
||
524
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
235 |
\begin{lemma} \label{lem:bc-btc} |
520 | 236 |
The inclusion $\bc_*(X) \sub \btc_*(X)$ is a homotopy equivalence |
237 |
\end{lemma} |
|
238 |
||
239 |
Before giving the proof we need a few preliminary results. |
|
514
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
240 |
|
521
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
241 |
\begin{lemma} \label{bt-contract} |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
242 |
$\btc_*(B^n)$ is contractible (acyclic in positive degrees). |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
243 |
\end{lemma} |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
244 |
\begin{proof} |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
245 |
We will construct a contracting homotopy $h: \btc_*(B^n)\to \btc_*(B^n)$. |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
246 |
|
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
247 |
We will assume a splitting $s:H_0(\btc_*(B^n))\to \btc_0(B^n)$ |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
248 |
of the quotient map $q:\btc_0(B^n)\to H_0(\btc_*(B^n))$. |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
249 |
Let $r = s\circ q$. |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
250 |
|
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
251 |
For $x\in \btc_{ij}$ with $i\ge 1$ define |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
252 |
\[ |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
253 |
h(x) = e(x) , |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
254 |
\] |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
255 |
where |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
256 |
\[ |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
257 |
e: \btc_{ij}\to\btc_{i+1,j} |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
258 |
\] |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
259 |
adds an outermost blob, equal to all of $B^n$, to the $j$-parameter family of blob diagrams. |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
260 |
|
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
261 |
A generator $y\in \btc_{0j}$ is a map $y:P\to \BD_0$, where $P$ is some $j$-dimensional polyhedron. |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
262 |
We define $r(y)\in \btc_{0j}$ to be the constant function $r\circ y : P\to \BD_0$. |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
263 |
Let $c(r(y))\in \btc_{0,j+1}$ be the constant map from the cone of $P$ to $\BD_0$ taking |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
264 |
the same value (i.e.\ $r(y(p))$ for any $p\in P$). |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
265 |
Let $e(y - r(y)) \in \btc_{1j}$ denote the $j$-parameter family of 1-blob diagrams |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
266 |
whose value at $p\in P$ is the blob $B^n$ with label $y(p) - r(y(p))$. |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
267 |
Now define, for $y\in \btc_{0j}$, |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
268 |
\[ |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
269 |
h(y) = e(y - r(y)) + c(r(y)) . |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
270 |
\] |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
271 |
\nn{up to sign, at least} |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
272 |
|
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
273 |
We must now verify that $h$ does the job it was intended to do. |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
274 |
For $x\in \btc_{ij}$ with $i\ge 2$ we have |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
275 |
\nn{ignoring signs} |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
276 |
\begin{align*} |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
277 |
\bd h(x) + h(\bd x) &= \bd(e(x)) + e(\bd x) \\ |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
278 |
&= \bd_b(e(x)) + \bd_t(e(x)) + e(\bd_b x) + e(\bd_t x) \\ |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
279 |
&= \bd_b(e(x)) + e(\bd_b x) \quad\quad\text{(since $\bd_t(e(x)) = e(\bd_t x)$)} \\ |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
280 |
&= x . |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
281 |
\end{align*} |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
282 |
For $x\in \btc_{1j}$ we have |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
283 |
\nn{ignoring signs} |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
284 |
\begin{align*} |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
285 |
\bd h(x) + h(\bd x) &= \bd_b(e(x)) + \bd_t(e(x)) + e(\bd_b x - r(\bd_b x)) + c(r(\bd_b x)) + e(\bd_t x) \\ |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
286 |
&= \bd_b(e(x)) + e(\bd_b x) \quad\quad\text{(since $r(\bd_b x) = 0$)} \\ |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
287 |
&= x . |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
288 |
\end{align*} |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
289 |
For $x\in \btc_{0j}$ with $j\ge 1$ we have |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
290 |
\nn{ignoring signs} |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
291 |
\begin{align*} |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
292 |
\bd h(x) + h(\bd x) &= \bd_b(e(x - r(x))) + \bd_t(e(x - r(x))) + \bd_t(c(r(x))) + |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
293 |
e(\bd_t x - r(\bd_t x)) + c(r(\bd_t x)) \\ |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
294 |
&= x - r(x) + \bd_t(c(r(x))) + c(r(\bd_t x)) \\ |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
295 |
&= x - r(x) + r(x) \\ |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
296 |
&= x. |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
297 |
\end{align*} |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
298 |
For $x\in \btc_{00}$ we have |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
299 |
\nn{ignoring signs} |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
300 |
\begin{align*} |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
301 |
\bd h(x) + h(\bd x) &= \bd_b(e(x - r(x))) + \bd_t(c(r(x))) \\ |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
302 |
&= x - r(x) + r(x) - r(x)\\ |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
303 |
&= x - r(x). |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
304 |
\end{align*} |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
305 |
\end{proof} |
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
306 |
|
524
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
307 |
\begin{lemma} \label{btc-prod} |
523 | 308 |
For manifolds $X$ and $Y$, we have $\btc_*(X\du Y) \simeq \btc_*(X)\otimes\btc_*(Y)$. |
309 |
\end{lemma} |
|
310 |
\begin{proof} |
|
311 |
This follows from the Eilenber-Zilber theorem and the fact that |
|
312 |
\[ |
|
313 |
\BD_k(X\du Y) \cong \coprod_{i+j=k} \BD_i(X)\times\BD_j(Y) . |
|
314 |
\] |
|
315 |
\end{proof} |
|
316 |
||
317 |
For $S\sub X$, we say that $a\in \btc_k(X)$ is {\it supported on $S$} |
|
318 |
if there exists $S' \subeq S$, $a'\in \btc_k(S')$ |
|
319 |
and $r\in \btc_0(X\setmin S')$ such that $a = a'\bullet r$. |
|
320 |
||
321 |
\newcommand\sbtc{\btc^{\cU}} |
|
322 |
Let $\cU$ be an open cover of $X$. |
|
323 |
Let $\sbtc_*(X)\sub\btc_*(X)$ be the subcomplex generated by |
|
324 |
$a\in \btc_*(X)$ such that there is a decomposition $X = \cup_i D_i$ |
|
325 |
such that each $D_i$ is a ball contained in some open set of $\cU$ and |
|
326 |
$a$ is splittable along this decomposition. |
|
327 |
In other words, $a$ can be obtained by gluing together pieces, each of which |
|
328 |
is small with respect to $\cU$. |
|
329 |
||
330 |
\begin{lemma} \label{small-top-blobs} |
|
331 |
For any open cover $\cU$ of $X$, the inclusion $\sbtc_*(X)\sub\btc_*(X)$ |
|
332 |
is a homotopy equivalence. |
|
333 |
\end{lemma} |
|
334 |
\begin{proof} |
|
335 |
This follows from a combination of Lemma \ref{extension_lemma_c} and the techniques of |
|
336 |
the proof of Lemma \ref{small-blobs-b}. |
|
337 |
||
338 |
It suffices to show that we can deform a finite subcomplex $C_*$ of $\btc_*(X)$ into $\sbtc_*(X)$ |
|
339 |
(relative to any designated subcomplex of $C_*$ already in $\sbtc_*(X)$). |
|
340 |
The first step is to replace families of general blob diagrams with families that are |
|
341 |
small with respect to $\cU$. |
|
342 |
This is done as in the proof of Lemma \ref{small-blobs-b}; the technique of the proof works in families. |
|
343 |
Each such family is homotopic to a sum families which can be a ``lifted" to $\Homeo(X)$. |
|
344 |
That is, $f:P \to \BD_k$ has the form $f(p) = g(p)(b)$ for some $g:P\to \Homeo(X)$ and $b\in \BD_k$. |
|
345 |
(We are ignoring a complication related to twig blob labels, which might vary |
|
346 |
independently of $g$, but this complication does not affect the conclusion we draw here.) |
|
347 |
We now apply Lemma \ref{extension_lemma_c} to get families which are supported |
|
348 |
on balls $D_i$ contained in open sets of $\cU$. |
|
349 |
\end{proof} |
|
350 |
||
351 |
||
524
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
352 |
\begin{proof}[Proof of \ref{lem:bc-btc}] |
523 | 353 |
Armed with the above lemmas, we can now proceed similarly to the proof of \ref{small-blobs-b}. |
354 |
||
355 |
It suffices to show that for any finitely generated pair of subcomplexes |
|
356 |
$(C_*, D_*) \sub (\btc_*(X), \bc_*(X))$ |
|
357 |
we can find a homotopy $h:C_*\to \btc_*(X)$ such that $h(D_*) \sub \bc_*(X)$ |
|
358 |
and $x + h\bd(x) + \bd h(X) \in \bc_*(X)$ for all $x\in C_*$. |
|
359 |
||
360 |
By Lemma \ref{small-top-blobs}, we may assume that $C_* \sub \btc_*^\cU(X)$ for some |
|
361 |
cover $\cU$ of our choosing. |
|
362 |
We choose $\cU$ fine enough so that each generator of $C_*$ is supported on a disjoint union of balls. |
|
363 |
(This is possible since the original $C_*$ was finite and therefore had bounded dimension.) |
|
364 |
||
365 |
Since $\bc_0(X) = \btc_0(X)$, we can take $h_0 = 0$. |
|
366 |
||
524
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
367 |
Let $b \in C_1$ be a generator. |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
368 |
Since $b$ is supported in a disjoint union of balls, |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
369 |
we can find $s(b)\in \bc_1$ with $\bd (s(b)) = \bd b$ |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
370 |
(by \ref{disj-union-contract}), and also $h_1(b) \in \btc_2$ |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
371 |
such that $\bd (h_1(b)) = s(b) - b$ |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
372 |
(by \ref{bt-contract} and \ref{btc-prod}). |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
373 |
|
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
374 |
Now let $b$ be a generator of $C_2$. |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
375 |
If $\cU$ is fine enough, there is a disjoint union of balls $V$ |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
376 |
on which $b + h_1(\bd b)$ is supported. |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
377 |
Since $\bd(b + h_1(\bd b)) = s(\bd b) \in \bc_2$, we can find |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
378 |
$s(b)\in \bc_2$ with $\bd(s(b)) = \bd(b + h_1(\bd b))$ (by \ref{disj-union-contract}). |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
379 |
By \ref{bt-contract} and \ref{btc-prod}, we can now find |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
380 |
$h_2(b) \in \btc_3$, also supported on $V$, such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$ |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
381 |
|
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
382 |
The general case, $h_k$, is similar. |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
383 |
\end{proof} |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
384 |
|
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
385 |
The proof of \ref{lem:bc-btc} constructs a homotopy inverse to the inclusion |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
386 |
$\bc_*(X)\sub \btc_*(X)$. |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
387 |
One might ask for more: a contractible set of possible homotopy inverses, or at least an |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
388 |
$m$-connected set for arbitrarily large $m$. |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
389 |
The latter can be achieved with finer control over the various |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
390 |
choices of disjoint unions of balls in the above proofs, but we will not pursue this here. |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
391 |
|
523 | 392 |
|
393 |
||
394 |
||
524
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
395 |
\subsection{Action of \texorpdfstring{$\CH{X}$}{C*(Homeo(M))}} |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
396 |
\label{ss:emap-def} |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
397 |
|
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
398 |
Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
399 |
the space of homeomorphisms |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
400 |
between the $n$-manifolds $X$ and $Y$ |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
401 |
(any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
402 |
We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
403 |
(For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
404 |
than simplices --- they can be based on any linear polyhedron. |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
405 |
\nn{be more restrictive here? does more need to be said?}) |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
406 |
|
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
407 |
\begin{thm} \label{thm:CH} |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
408 |
For $n$-manifolds $X$ and $Y$ there is a chain map |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
409 |
\eq{ |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
410 |
e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) , |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
411 |
} |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
412 |
well-defined up to homotopy, |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
413 |
such that |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
414 |
\begin{enumerate} |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
415 |
\item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
416 |
$\Homeo(X, Y)$ on $\bc_*(X)$ described in Property (\ref{property:functoriality}), and |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
417 |
\item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
418 |
the following diagram commutes up to homotopy |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
419 |
\begin{equation*} |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
420 |
\xymatrix@C+2cm{ |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
421 |
CH_*(X, Y) \otimes \bc_*(X) |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
422 |
\ar[r]_(.6){e_{XY}} \ar[d]^{\gl \otimes \gl} & |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
423 |
\bc_*(Y)\ar[d]^{\gl} \\ |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
424 |
CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]_(.6){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
425 |
} |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
426 |
\end{equation*} |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
427 |
\end{enumerate} |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
428 |
\end{thm} |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
429 |
|
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
430 |
\begin{proof} |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
431 |
In light of Lemma \ref{lem:bc-btc}, it suffices to prove the theorem with |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
432 |
$\bc_*$ replaced by $\btc_*$. |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
433 |
And in fact for $\btc_*$ we get a sharper result: we can omit |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
434 |
the ``up to homotopy" qualifiers. |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
435 |
|
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
436 |
Let $f\in CH_k(X, Y)$, $f:P^k\to \Homeo(X \to Y)$ and $a\in \btc_{ij}(X)$, |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
437 |
$a:Q^j \to \BD_i(X)$. |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
438 |
Define $e_{XY}(f\ot a)\in \btc_{i,j+k}(Y)$ by |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
439 |
\begin{align*} |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
440 |
e_{XY}(f\ot a) : P\times Q &\to \BD_i(Y) \\ |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
441 |
(p,q) &\mapsto f(p)(a(q)) . |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
442 |
\end{align*} |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
443 |
It is clear that this agrees with the previously defined $CH_0(X, Y)$ action on $\btc_*$, |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
444 |
and it is also easy to see that the diagram in item 2 of the statement of the theorem |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
445 |
commutes on the nose. |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
446 |
\end{proof} |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
447 |
|
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
448 |
|
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
449 |
\begin{thm} |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
450 |
\label{thm:CH-associativity} |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
451 |
The $CH_*(X, Y)$ actions defined above are associative. |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
452 |
That is, the following diagram commutes up to homotopy: |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
453 |
\[ \xymatrix{ |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
454 |
& CH_*(Y, Z) \ot \bc_*(Y) \ar[dr]^{e_{YZ}} & \\ |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
455 |
CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & \bc_*(Z) \\ |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
456 |
& CH_*(X, Z) \ot \bc_*(X) \ar[ur]_{e_{XZ}} & |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
457 |
} \] |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
458 |
Here $\mu:CH_*(X, Y) \ot CH_*(Y, Z)\to CH_*(X, Z)$ is the map induced by composition |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
459 |
of homeomorphisms. |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
460 |
\end{thm} |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
461 |
\begin{proof} |
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
462 |
The corresponding diagram for $\btc_*$ commutes on the nose. |
523 | 463 |
\end{proof} |
464 |
||
465 |
||
521
4a988e00468a
local contractibility for SOSS blob complex
Kevin Walker <kevin@canyon23.net>
parents:
520
diff
changeset
|
466 |
|
514
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
467 |
|
bb696f417f22
starting yet again on evmap
Kevin Walker <kevin@canyon23.net>
parents:
513
diff
changeset
|
468 |
|
520 | 469 |
|
470 |
||
524
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
471 |
\noop{ |
512
050dba5e7bdd
fixing some (but not all!?) of the hyperref warnings; start on revision of evmap
Kevin Walker <kevin@canyon23.net>
parents:
494
diff
changeset
|
472 |
|
050dba5e7bdd
fixing some (but not all!?) of the hyperref warnings; start on revision of evmap
Kevin Walker <kevin@canyon23.net>
parents:
494
diff
changeset
|
473 |
|
050dba5e7bdd
fixing some (but not all!?) of the hyperref warnings; start on revision of evmap
Kevin Walker <kevin@canyon23.net>
parents:
494
diff
changeset
|
474 |
\subsection{[older version still hanging around]} |
513 | 475 |
\label{ss:old-evmap-remnants} |
512
050dba5e7bdd
fixing some (but not all!?) of the hyperref warnings; start on revision of evmap
Kevin Walker <kevin@canyon23.net>
parents:
494
diff
changeset
|
476 |
|
246 | 477 |
\nn{should comment at the start about any assumptions about smooth, PL etc.} |
478 |
||
447 | 479 |
\nn{should maybe mention alternate def of blob complex (sort-of-simplicial space instead of |
480 |
sort-of-simplicial set) where this action would be easy} |
|
481 |
||
236 | 482 |
Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of |
483 |
the space of homeomorphisms |
|
430 | 484 |
between the $n$-manifolds $X$ and $Y$ (any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). |
249 | 485 |
We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
486 |
(For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
|
70 | 487 |
than simplices --- they can be based on any linear polyhedron. |
249 | 488 |
\nn{be more restrictive here? does more need to be said?}) |
70 | 489 |
|
437 | 490 |
\begin{thm} \label{thm:CH} |
70 | 491 |
For $n$-manifolds $X$ and $Y$ there is a chain map |
492 |
\eq{ |
|
244
cf01e213044a
start working on "evaluation map" section
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
493 |
e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) |
70 | 494 |
} |
244
cf01e213044a
start working on "evaluation map" section
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
495 |
such that |
cf01e213044a
start working on "evaluation map" section
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
496 |
\begin{enumerate} |
cf01e213044a
start working on "evaluation map" section
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
497 |
\item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of |
437 | 498 |
$\Homeo(X, Y)$ on $\bc_*(X)$ described in Property (\ref{property:functoriality}), and |
244
cf01e213044a
start working on "evaluation map" section
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
499 |
\item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
70 | 500 |
the following diagram commutes up to homotopy |
430 | 501 |
\begin{equation*} |
502 |
\xymatrix@C+2cm{ |
|
236 | 503 |
CH_*(X, Y) \otimes \bc_*(X) |
430 | 504 |
\ar[r]_(.6){e_{XY}} \ar[d]^{\gl \otimes \gl} & |
505 |
\bc_*(Y)\ar[d]^{\gl} \\ |
|
506 |
CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]_(.6){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) |
|
507 |
} |
|
508 |
\end{equation*} |
|
244
cf01e213044a
start working on "evaluation map" section
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
509 |
\end{enumerate} |
453
e88e44347b36
weaking thm:CH for iterated homotopy
Scott Morrison <scott@tqft.net>
parents:
447
diff
changeset
|
510 |
Moreover, for any $m \geq 0$, we can find a family of chain maps $\{e_{XY}\}$ |
e88e44347b36
weaking thm:CH for iterated homotopy
Scott Morrison <scott@tqft.net>
parents:
447
diff
changeset
|
511 |
satisfying the above two conditions which is $m$-connected. In particular, this means that the choice of chain map above is unique up to homotopy. |
437 | 512 |
\end{thm} |
453
e88e44347b36
weaking thm:CH for iterated homotopy
Scott Morrison <scott@tqft.net>
parents:
447
diff
changeset
|
513 |
\begin{rem} |
e88e44347b36
weaking thm:CH for iterated homotopy
Scott Morrison <scott@tqft.net>
parents:
447
diff
changeset
|
514 |
Note that the statement doesn't quite give uniqueness up to iterated homotopy. We fully expect that this should actually be the case, but haven't been able to prove this. |
e88e44347b36
weaking thm:CH for iterated homotopy
Scott Morrison <scott@tqft.net>
parents:
447
diff
changeset
|
515 |
\end{rem} |
e88e44347b36
weaking thm:CH for iterated homotopy
Scott Morrison <scott@tqft.net>
parents:
447
diff
changeset
|
516 |
|
70 | 517 |
|
345 | 518 |
Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma}, |
519 |
and then give an outline of the method of proof. |
|
303
2252c53bd449
minor changes in a few places
Scott Morrison <scott@tqft.net>
parents:
256
diff
changeset
|
520 |
|
70 | 521 |
Without loss of generality, we will assume $X = Y$. |
522 |
||
523 |
\medskip |
|
524 |
||
244
cf01e213044a
start working on "evaluation map" section
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
525 |
Let $f: P \times X \to X$ be a family of homeomorphisms (e.g. a generator of $CH_*(X)$) |
cf01e213044a
start working on "evaluation map" section
Kevin Walker <kevin@canyon23.net>
parents:
236
diff
changeset
|
526 |
and let $S \sub X$. |
70 | 527 |
We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all |
345 | 528 |
$x \notin S$ and $p, q \in P$. Equivalently, $f$ is supported on $S$ if |
417
d3b05641e7ca
making quotation marks consistently "American style"
Kevin Walker <kevin@canyon23.net>
parents:
415
diff
changeset
|
529 |
there is a family of homeomorphisms $f' : P \times S \to S$ and a ``background" |
236 | 530 |
homeomorphism $f_0 : X \to X$ so that |
245
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
244
diff
changeset
|
531 |
\begin{align*} |
70 | 532 |
f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\ |
533 |
\intertext{and} |
|
534 |
f(p,x) & = f_0(x) \;\;\;\; \mbox{for}\; (p, x) \in {P \times (X \setmin S)}. |
|
245
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
244
diff
changeset
|
535 |
\end{align*} |
70 | 536 |
Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$. |
245
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
244
diff
changeset
|
537 |
(So when we talk about ``the" support of a family, there is some ambiguity, |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
244
diff
changeset
|
538 |
but this ambiguity will not matter to us.) |
70 | 539 |
|
540 |
Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
|
236 | 541 |
A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is |
245
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
244
diff
changeset
|
542 |
{\it adapted to $\cU$} |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
244
diff
changeset
|
543 |
if the support of $f$ is contained in the union |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
244
diff
changeset
|
544 |
of at most $k$ of the $U_\alpha$'s. |
70 | 545 |
|
546 |
\begin{lemma} \label{extension_lemma} |
|
236 | 547 |
Let $x \in CH_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
548 |
Then $x$ is homotopic (rel boundary) to some $x' \in CH_k(X)$ which is adapted to $\cU$. |
|
70 | 549 |
Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
550 |
\end{lemma} |
|
551 |
||
426
8aca80203f9d
search & replace: s/((sub?)section|appendix)\s+\\ref/\S\ref/
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
552 |
The proof will be given in \S\ref{sec:localising}. |
70 | 553 |
|
554 |
\medskip |
|
555 |
||
437 | 556 |
Before diving into the details, we outline our strategy for the proof of Theorem \ref{thm:CH}. |
236 | 557 |
Let $p$ be a singular cell in $CH_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
246 | 558 |
We say that $p\ot b$ is {\it localizable} if there exists $V \sub X$ such that |
559 |
\begin{itemize} |
|
70 | 560 |
\item $V$ is homeomorphic to a disjoint union of balls, and |
561 |
\item $\supp(p) \cup \supp(b) \sub V$. |
|
246 | 562 |
\end{itemize} |
245
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
244
diff
changeset
|
563 |
(Recall that $\supp(b)$ is defined to be the union of the blobs of the diagram $b$.) |
246 | 564 |
|
565 |
Assuming that $p\ot b$ is localizable as above, |
|
566 |
let $W = X \setmin V$, $W' = p(W)$ and $V' = X\setmin W'$. |
|
70 | 567 |
We then have a factorization |
568 |
\[ |
|
569 |
p = \gl(q, r), |
|
570 |
\] |
|
236 | 571 |
where $q \in CH_k(V, V')$ and $r \in CH_0(W, W')$. |
73 | 572 |
We can also factorize $b = \gl(b_V, b_W)$, where $b_V\in \bc_*(V)$ and $b_W\in\bc_0(W)$. |
70 | 573 |
According to the commutative diagram of the proposition, we must have |
574 |
\[ |
|
73 | 575 |
e_X(p\otimes b) = e_X(\gl(q\otimes b_V, r\otimes b_W)) = |
576 |
gl(e_{VV'}(q\otimes b_V), e_{WW'}(r\otimes b_W)) . |
|
577 |
\] |
|
430 | 578 |
Since $r$ is a 0-parameter family of homeomorphisms, we must have |
73 | 579 |
\[ |
580 |
e_{WW'}(r\otimes b_W) = r(b_W), |
|
70 | 581 |
\] |
236 | 582 |
where $r(b_W)$ denotes the obvious action of homeomorphisms on blob diagrams (in |
73 | 583 |
this case a 0-blob diagram). |
584 |
Since $V'$ is a disjoint union of balls, $\bc_*(V')$ is acyclic in degrees $>0$ |
|
303
2252c53bd449
minor changes in a few places
Scott Morrison <scott@tqft.net>
parents:
256
diff
changeset
|
585 |
(by Properties \ref{property:disjoint-union} and \ref{property:contractibility}). |
73 | 586 |
Assuming inductively that we have already defined $e_{VV'}(\bd(q\otimes b_V))$, |
83 | 587 |
there is, up to (iterated) homotopy, a unique choice for $e_{VV'}(q\otimes b_V)$ |
73 | 588 |
such that |
589 |
\[ |
|
590 |
\bd(e_{VV'}(q\otimes b_V)) = e_{VV'}(\bd(q\otimes b_V)) . |
|
591 |
\] |
|
592 |
||
593 |
Thus the conditions of the proposition determine (up to homotopy) the evaluation |
|
246 | 594 |
map for localizable generators $p\otimes b$. |
73 | 595 |
On the other hand, Lemma \ref{extension_lemma} allows us to homotope |
246 | 596 |
arbitrary generators to sums of localizable generators. |
73 | 597 |
This (roughly) establishes the uniqueness part of the proposition. |
598 |
To show existence, we must show that the various choices involved in constructing |
|
599 |
evaluation maps in this way affect the final answer only by a homotopy. |
|
600 |
||
246 | 601 |
Now for a little more detail. |
602 |
(But we're still just motivating the full, gory details, which will follow.) |
|
434 | 603 |
Choose a metric on $X$, and let $\cU_\gamma$ be the open cover of $X$ by balls of radius $\gamma$. |
246 | 604 |
By Lemma \ref{extension_lemma} we can restrict our attention to $k$-parameter families |
605 |
$p$ of homeomorphisms such that $\supp(p)$ is contained in the union of $k$ $\gamma$-balls. |
|
606 |
For fixed blob diagram $b$ and fixed $k$, it's not hard to show that for $\gamma$ small enough |
|
607 |
$p\ot b$ must be localizable. |
|
608 |
On the other hand, for fixed $k$ and $\gamma$ there exist $p$ and $b$ such that $p\ot b$ is not localizable, |
|
609 |
and for fixed $\gamma$ and $b$ there exist non-localizable $p\ot b$ for sufficiently large $k$. |
|
610 |
Thus we will need to take an appropriate limit as $\gamma$ approaches zero. |
|
83 | 611 |
|
246 | 612 |
The construction of $e_X$, as outlined above, depends on various choices, one of which |
613 |
is the choice, for each localizable generator $p\ot b$, |
|
614 |
of disjoint balls $V$ containing $\supp(p)\cup\supp(b)$. |
|
615 |
Let $V'$ be another disjoint union of balls containing $\supp(p)\cup\supp(b)$, |
|
430 | 616 |
and assume that there exists yet another disjoint union of balls $W$ containing |
246 | 617 |
$V\cup V'$. |
618 |
Then we can use $W$ to construct a homotopy between the two versions of $e_X$ |
|
619 |
associated to $V$ and $V'$. |
|
620 |
If we impose no constraints on $V$ and $V'$ then such a $W$ need not exist. |
|
621 |
Thus we will insist below that $V$ (and $V'$) be contained in small metric neighborhoods |
|
622 |
of $\supp(p)\cup\supp(b)$. |
|
623 |
Because we want not mere homotopy uniqueness but iterated homotopy uniqueness, |
|
624 |
we will similarly require that $W$ be contained in a slightly larger metric neighborhood of |
|
625 |
$\supp(p)\cup\supp(b)$, and so on. |
|
626 |
||
83 | 627 |
|
437 | 628 |
\begin{proof}[Proof of Theorem \ref{thm:CH}.] |
430 | 629 |
We'll use the notation $|b| = \supp(b)$ and $|p| = \supp(p)$. |
83 | 630 |
|
631 |
Choose a metric on $X$. |
|
434 | 632 |
Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero |
83 | 633 |
(e.g.\ $\ep_i = 2^{-i}$). |
634 |
Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$ |
|
635 |
converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$). |
|
88 | 636 |
Let $\phi_l$ be an increasing sequence of positive numbers |
430 | 637 |
satisfying the inequalities of Lemma \ref{xx2phi} below. |
236 | 638 |
Given a generator $p\otimes b$ of $CH_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $l$ |
83 | 639 |
define |
640 |
\[ |
|
88 | 641 |
N_{i,l}(p\ot b) \deq \Nbd_{l\ep_i}(|b|) \cup \Nbd_{\phi_l\delta_i}(|p|). |
83 | 642 |
\] |
247 | 643 |
In other words, for each $i$ |
644 |
we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized |
|
88 | 645 |
by $l$), with $\ep_i$ controlling the size of the buffers around $|b|$ and $\delta_i$ controlling |
646 |
the size of the buffers around $|p|$. |
|
83 | 647 |
|
236 | 648 |
Next we define subcomplexes $G_*^{i,m} \sub CH_*(X)\otimes \bc_*(X)$. |
649 |
Let $p\ot b$ be a generator of $CH_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b) |
|
83 | 650 |
= \deg(p) + \deg(b)$. |
430 | 651 |
We say $p\ot b$ is in $G_*^{i,m}$ exactly when either (a) $\deg(p) = 0$ or (b) |
84 | 652 |
there exist codimension-zero submanifolds $V_0,\ldots,V_m \sub X$ such that each $V_j$ |
83 | 653 |
is homeomorphic to a disjoint union of balls and |
654 |
\[ |
|
84 | 655 |
N_{i,k}(p\ot b) \subeq V_0 \subeq N_{i,k+1}(p\ot b) |
434 | 656 |
\subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,k+m+1}(p\ot b) , |
83 | 657 |
\] |
430 | 658 |
and further $\bd(p\ot b) \in G_*^{i,m}$. |
83 | 659 |
We also require that $b$ is splitable (transverse) along the boundary of each $V_l$. |
660 |
||
661 |
Note that $G_*^{i,m+1} \subeq G_*^{i,m}$. |
|
73 | 662 |
|
83 | 663 |
As sketched above and explained in detail below, |
664 |
$G_*^{i,m}$ is a subcomplex where it is easy to define |
|
665 |
the evaluation map. |
|
84 | 666 |
The parameter $m$ controls the number of iterated homotopies we are able to construct |
87 | 667 |
(see Lemma \ref{m_order_hty}). |
83 | 668 |
The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of |
236 | 669 |
$CH_*(X)\ot \bc_*(X)$ (see Lemma \ref{Gim_approx}). |
83 | 670 |
|
249 | 671 |
Next we define a chain map (dependent on some choices) $e_{i,m}: G_*^{i,m} \to \bc_*(X)$. |
672 |
(When the domain is clear from context we will drop the subscripts and write |
|
673 |
simply $e: G_*^{i,m} \to \bc_*(X)$). |
|
83 | 674 |
Let $p\ot b \in G_*^{i,m}$. |
675 |
If $\deg(p) = 0$, define |
|
676 |
\[ |
|
677 |
e(p\ot b) = p(b) , |
|
678 |
\] |
|
236 | 679 |
where $p(b)$ denotes the obvious action of the homeomorphism(s) $p$ on the blob diagram $b$. |
83 | 680 |
For general $p\ot b$ ($\deg(p) \ge 1$) assume inductively that we have already defined |
681 |
$e(p'\ot b')$ when $\deg(p') + \deg(b') < k = \deg(p) + \deg(b)$. |
|
84 | 682 |
Choose $V = V_0$ as above so that |
83 | 683 |
\[ |
84 | 684 |
N_{i,k}(p\ot b) \subeq V \subeq N_{i,k+1}(p\ot b) . |
83 | 685 |
\] |
84 | 686 |
Let $\bd(p\ot b) = \sum_j p_j\ot b_j$, and let $V^j$ be the choice of neighborhood |
83 | 687 |
of $|p_j|\cup |b_j|$ made at the preceding stage of the induction. |
688 |
For all $j$, |
|
689 |
\[ |
|
88 | 690 |
V^j \subeq N_{i,k}(p_j\ot b_j) \subeq N_{i,k}(p\ot b) \subeq V . |
83 | 691 |
\] |
692 |
(The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.) |
|
693 |
We therefore have splittings |
|
694 |
\[ |
|
247 | 695 |
p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet f'' , |
83 | 696 |
\] |
236 | 697 |
where $p' \in CH_*(V)$, $p'' \in CH_*(X\setmin V)$, |
84 | 698 |
$b' \in \bc_*(V)$, $b'' \in \bc_*(X\setmin V)$, |
86 | 699 |
$f' \in \bc_*(p(V))$, and $f'' \in \bc_*(p(X\setmin V))$. |
236 | 700 |
(Note that since the family of homeomorphisms $p$ is constant (independent of parameters) |
86 | 701 |
near $\bd V$, the expressions $p(V) \sub X$ and $p(X\setmin V) \sub X$ are |
83 | 702 |
unambiguous.) |
86 | 703 |
We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$. |
704 |
%We also have that $\deg(b'') = 0 = \deg(p'')$. |
|
84 | 705 |
Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$. |
402 | 706 |
This is possible by Properties \ref{property:disjoint-union} and \ref{property:contractibility} and the fact that isotopic fields |
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
413
diff
changeset
|
707 |
differ by a local relation. |
83 | 708 |
Finally, define |
709 |
\[ |
|
710 |
e(p\ot b) \deq x' \bullet p''(b'') . |
|
711 |
\] |
|
73 | 712 |
|
492
833bd74143a4
put in a stub appendix for MoAM, but I'm going to go do other things next
Scott Morrison <scott@tqft.net>
parents:
453
diff
changeset
|
713 |
Note that above we are essentially using the method of acyclic models \nn{\S \ref{sec:moam}}. |
84 | 714 |
For each generator $p\ot b$ we specify the acyclic (in positive degrees) |
715 |
target complex $\bc_*(p(V)) \bullet p''(b'')$. |
|
716 |
||
717 |
The definition of $e: G_*^{i,m} \to \bc_*(X)$ depends on two sets of choices: |
|
718 |
The choice of neighborhoods $V$ and the choice of inverse boundaries $x'$. |
|
88 | 719 |
The next lemma shows that up to (iterated) homotopy $e$ is independent |
84 | 720 |
of these choices. |
88 | 721 |
(Note that independence of choices of $x'$ (for fixed choices of $V$) |
722 |
is a standard result in the method of acyclic models.) |
|
84 | 723 |
|
88 | 724 |
%\begin{lemma} |
725 |
%Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with |
|
726 |
%different choices of $x'$ at each step. |
|
727 |
%(Same choice of $V$ at each step.) |
|
728 |
%Then $e$ and $\tilde{e}$ are homotopic via a homotopy in $\bc_*(p(V)) \bullet p''(b'')$. |
|
729 |
%Any two choices of such a first-order homotopy are second-order homotopic, and so on, |
|
730 |
%to arbitrary order. |
|
731 |
%\end{lemma} |
|
84 | 732 |
|
88 | 733 |
%\begin{proof} |
734 |
%This is a standard result in the method of acyclic models. |
|
735 |
%\nn{should we say more here?} |
|
736 |
%\nn{maybe this lemma should be subsumed into the next lemma. probably it should.} |
|
737 |
%\end{proof} |
|
84 | 738 |
|
87 | 739 |
\begin{lemma} \label{m_order_hty} |
84 | 740 |
Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with |
741 |
different choices of $V$ (and hence also different choices of $x'$) at each step. |
|
742 |
If $m \ge 1$ then $e$ and $\tilde{e}$ are homotopic. |
|
743 |
If $m \ge 2$ then any two choices of this first-order homotopy are second-order homotopic. |
|
430 | 744 |
Continuing, $e : G_*^{i,m} \to \bc_*(X)$ is well-defined up to $m$-th order homotopy. |
84 | 745 |
\end{lemma} |
746 |
||
747 |
\begin{proof} |
|
748 |
We construct $h: G_*^{i,m} \to \bc_*(X)$ such that $\bd h + h\bd = e - \tilde{e}$. |
|
430 | 749 |
The chain maps $e$ and $\tilde{e}$ coincide on bidegrees $(0, j)$, so define $h$ |
84 | 750 |
to be zero there. |
751 |
Assume inductively that $h$ has been defined for degrees less than $k$. |
|
752 |
Let $p\ot b$ be a generator of degree $k$. |
|
753 |
Choose $V_1$ as in the definition of $G_*^{i,m}$ so that |
|
754 |
\[ |
|
755 |
N_{i,k+1}(p\ot b) \subeq V_1 \subeq N_{i,k+2}(p\ot b) . |
|
756 |
\] |
|
757 |
There are splittings |
|
758 |
\[ |
|
759 |
p = p'_1\bullet p''_1 , \;\; b = b'_1\bullet b''_1 , |
|
760 |
\;\; e(p\ot b) - \tilde{e}(p\ot b) - h(\bd(p\ot b)) = f'_1\bullet f''_1 , |
|
761 |
\] |
|
236 | 762 |
where $p'_1 \in CH_*(V_1)$, $p''_1 \in CH_*(X\setmin V_1)$, |
84 | 763 |
$b'_1 \in \bc_*(V_1)$, $b''_1 \in \bc_*(X\setmin V_1)$, |
764 |
$f'_1 \in \bc_*(p(V_1))$, and $f''_1 \in \bc_*(p(X\setmin V_1))$. |
|
88 | 765 |
Inductively, $\bd f'_1 = 0$ and $f_1'' = p_1''(b_1'')$. |
84 | 766 |
Choose $x'_1 \in \bc_*(p(V_1))$ so that $\bd x'_1 = f'_1$. |
767 |
Define |
|
768 |
\[ |
|
769 |
h(p\ot b) \deq x'_1 \bullet p''_1(b''_1) . |
|
770 |
\] |
|
771 |
This completes the construction of the first-order homotopy when $m \ge 1$. |
|
772 |
||
773 |
The $j$-th order homotopy is constructed similarly, with $V_j$ replacing $V_1$ above. |
|
774 |
\end{proof} |
|
775 |
||
776 |
Note that on $G_*^{i,m+1} \subeq G_*^{i,m}$, we have defined two maps, |
|
249 | 777 |
$e_{i,m}$ and $e_{i,m+1}$. |
778 |
An easy variation on the above lemma shows that |
|
779 |
the restrictions of $e_{i,m}$ and $e_{i,m+1}$ to $G_*^{i,m+1}$ are $m$-th |
|
84 | 780 |
order homotopic. |
781 |
||
236 | 782 |
Next we show how to homotope chains in $CH_*(X)\ot \bc_*(X)$ to one of the |
85 | 783 |
$G_*^{i,m}$. |
784 |
Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero. |
|
785 |
Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$. |
|
345 | 786 |
Let $h_j: CH_*(X)\to CH_*(X)$ be a chain map homotopic to the identity whose image is |
787 |
spanned by families of homeomorphisms with support compatible with $\cU_j$, |
|
788 |
as described in Lemma \ref{extension_lemma}. |
|
86 | 789 |
Recall that $h_j$ and also the homotopy connecting it to the identity do not increase |
85 | 790 |
supports. |
791 |
Define |
|
792 |
\[ |
|
793 |
g_j \deq h_j\circ h_{j-1} \circ \cdots \circ h_1 . |
|
794 |
\] |
|
795 |
The next lemma says that for all generators $p\ot b$ we can choose $j$ large enough so that |
|
796 |
$g_j(p)\ot b$ lies in $G_*^{i,m}$, for arbitrary $m$ and sufficiently large $i$ |
|
247 | 797 |
(depending on $b$, $\deg(p)$ and $m$). |
798 |
%(Note: Don't confuse this $n$ with the top dimension $n$ used elsewhere in this paper.) |
|
85 | 799 |
|
87 | 800 |
\begin{lemma} \label{Gim_approx} |
236 | 801 |
Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CH_*(X)$. |
85 | 802 |
Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$ |
255 | 803 |
there exists another constant $j_{ibmn}$ such that for all $j \ge j_{ibmn}$ and all $p\in CH_n(X)$ |
85 | 804 |
we have $g_j(p)\ot b \in G_*^{i,m}$. |
805 |
\end{lemma} |
|
806 |
||
255 | 807 |
For convenience we also define $k_{bmp} = k_{bmn}$ |
808 |
and $j_{ibmp} = j_{ibmn}$ where $n=\deg(p)$. |
|
254 | 809 |
Note that we may assume that |
810 |
\[ |
|
811 |
k_{bmp} \ge k_{alq} |
|
812 |
\] |
|
813 |
for all $l\ge m$ and all $q\ot a$ which appear in the boundary of $p\ot b$. |
|
255 | 814 |
Additionally, we may assume that |
815 |
\[ |
|
816 |
j_{ibmp} \ge j_{ialq} |
|
817 |
\] |
|
818 |
for all $l\ge m$ and all $q\ot a$ which appear in the boundary of $p\ot b$. |
|
819 |
||
254 | 820 |
|
85 | 821 |
\begin{proof} |
430 | 822 |
|
453
e88e44347b36
weaking thm:CH for iterated homotopy
Scott Morrison <scott@tqft.net>
parents:
447
diff
changeset
|
823 |
There exists $\lambda > 0$ such that for every subset $c$ of the blobs of $b$ the set $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ . |
434 | 824 |
(Here we are using the fact that the blobs are |
825 |
piecewise smooth or piecewise-linear and that $\bd c$ is collared.) |
|
90 | 826 |
We need to consider all such $c$ because all generators appearing in |
247 | 827 |
iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.) |
85 | 828 |
|
86 | 829 |
Let $r = \deg(b)$ and |
830 |
\[ |
|
90 | 831 |
t = r+n+m+1 = \deg(p\ot b) + m + 1. |
86 | 832 |
\] |
85 | 833 |
|
834 |
Choose $k = k_{bmn}$ such that |
|
835 |
\[ |
|
248 | 836 |
t\ep_k < \lambda |
85 | 837 |
\] |
838 |
and |
|
839 |
\[ |
|
90 | 840 |
n\cdot (2 (\phi_t + 1) \delta_k) < \ep_k . |
85 | 841 |
\] |
842 |
Let $i \ge k_{bmn}$. |
|
843 |
Choose $j = j_i$ so that |
|
844 |
\[ |
|
90 | 845 |
\gamma_j < \delta_i |
846 |
\] |
|
847 |
and also so that $\phi_t \gamma_j$ is less than the constant $\rho(M)$ of Lemma \ref{xxzz11}. |
|
848 |
||
236 | 849 |
Let $j \ge j_i$ and $p\in CH_n(X)$. |
90 | 850 |
Let $q$ be a generator appearing in $g_j(p)$. |
851 |
Note that $|q|$ is contained in a union of $n$ elements of the cover $\cU_j$, |
|
852 |
which implies that $|q|$ is contained in a union of $n$ metric balls of radius $\delta_i$. |
|
853 |
We must show that $q\ot b \in G_*^{i,m}$, which means finding neighborhoods |
|
854 |
$V_0,\ldots,V_m \sub X$ of $|q|\cup |b|$ such that each $V_j$ |
|
855 |
is homeomorphic to a disjoint union of balls and |
|
856 |
\[ |
|
857 |
N_{i,n}(q\ot b) \subeq V_0 \subeq N_{i,n+1}(q\ot b) |
|
858 |
\subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,t}(q\ot b) . |
|
859 |
\] |
|
248 | 860 |
Recall that |
861 |
\[ |
|
862 |
N_{i,a}(q\ot b) \deq \Nbd_{a\ep_i}(|b|) \cup \Nbd_{\phi_a\delta_i}(|q|). |
|
863 |
\] |
|
90 | 864 |
By repeated applications of Lemma \ref{xx2phi} we can find neighborhoods $U_0,\ldots,U_m$ |
865 |
of $|q|$, each homeomorphic to a disjoint union of balls, with |
|
866 |
\[ |
|
867 |
\Nbd_{\phi_{n+l} \delta_i}(|q|) \subeq U_l \subeq \Nbd_{\phi_{n+l+1} \delta_i}(|q|) . |
|
85 | 868 |
\] |
248 | 869 |
The inequalities above guarantee that |
870 |
for each $0\le l\le m$ we can find $u_l$ with |
|
90 | 871 |
\[ |
872 |
(n+l)\ep_i \le u_l \le (n+l+1)\ep_i |
|
873 |
\] |
|
874 |
such that each component of $U_l$ is either disjoint from $\Nbd_{u_l}(|b|)$ or contained in |
|
875 |
$\Nbd_{u_l}(|b|)$. |
|
876 |
This is because there are at most $n$ components of $U_l$, and each component |
|
877 |
has radius $\le (\phi_t + 1) \delta_i$. |
|
878 |
It follows that |
|
879 |
\[ |
|
880 |
V_l \deq \Nbd_{u_l}(|b|) \cup U_l |
|
881 |
\] |
|
882 |
is homeomorphic to a disjoint union of balls and satisfies |
|
883 |
\[ |
|
884 |
N_{i,n+l}(q\ot b) \subeq V_l \subeq N_{i,n+l+1}(q\ot b) . |
|
885 |
\] |
|
86 | 886 |
|
90 | 887 |
The same argument shows that each generator involved in iterated boundaries of $q\ot b$ |
888 |
is in $G_*^{i,m}$. |
|
86 | 889 |
\end{proof} |
890 |
||
430 | 891 |
In the next three lemmas, which provide the estimates needed above, we have made no effort to optimize the various bounds. |
86 | 892 |
(The bounds are, however, optimal in the sense of minimizing the amount of work |
893 |
we do. Equivalently, they are the first bounds we thought of.) |
|
894 |
||
895 |
We say that a subset $S$ of a metric space has radius $\le r$ if $S$ is contained in |
|
896 |
some metric ball of radius $r$. |
|
897 |
||
898 |
\begin{lemma} |
|
899 |
Let $S \sub \ebb^n$ (Euclidean $n$-space) have radius $\le r$. |
|
900 |
Then $\Nbd_a(S)$ is homeomorphic to a ball for $a \ge 2r$. |
|
901 |
\end{lemma} |
|
902 |
||
903 |
\begin{proof} \label{xxyy2} |
|
904 |
Let $S$ be contained in $B_r(y)$, $y \in \ebb^n$. |
|
89 | 905 |
Note that if $a \ge 2r$ then $\Nbd_a(S) \sup B_r(y)$. |
906 |
Let $z\in \Nbd_a(S) \setmin B_r(y)$. |
|
907 |
Consider the triangle |
|
494
cb76847c439e
many small fixes in ncat.tex
Scott Morrison <scott@tqft.net>
parents:
492
diff
changeset
|
908 |
with vertices $z$, $y$ and $s$ with $s\in S$ such that $z \in B_a(s)$. |
89 | 909 |
The length of the edge $yz$ is greater than $r$ which is greater |
910 |
than the length of the edge $ys$. |
|
911 |
It follows that the angle at $z$ is less than $\pi/2$ (less than $\pi/3$, in fact), |
|
912 |
which means that points on the edge $yz$ near $z$ are closer to $s$ than $z$ is, |
|
913 |
which implies that these points are also in $\Nbd_a(S)$. |
|
914 |
Hence $\Nbd_a(S)$ is star-shaped with respect to $y$. |
|
915 |
\end{proof} |
|
916 |
||
917 |
If we replace $\ebb^n$ above with an arbitrary compact Riemannian manifold $M$, |
|
918 |
the same result holds, so long as $a$ is not too large: |
|
430 | 919 |
\nn{replace this with a PL version} |
89 | 920 |
|
921 |
\begin{lemma} \label{xxzz11} |
|
922 |
Let $M$ be a compact Riemannian manifold. |
|
923 |
Then there is a constant $\rho(M)$ such that for all |
|
924 |
subsets $S\sub M$ of radius $\le r$ and all $a$ such that $2r \le a \le \rho(M)$, |
|
925 |
$\Nbd_a(S)$ is homeomorphic to a ball. |
|
926 |
\end{lemma} |
|
927 |
||
928 |
\begin{proof} |
|
929 |
Choose $\rho = \rho(M)$ such that $3\rho/2$ is less than the radius of injectivity of $M$, |
|
930 |
and also so that for any point $y\in M$ the geodesic coordinates of radius $3\rho/2$ around |
|
931 |
$y$ distort angles by only a small amount. |
|
932 |
Now the argument of the previous lemma works. |
|
85 | 933 |
\end{proof} |
934 |
||
935 |
||
89 | 936 |
|
937 |
\begin{lemma} \label{xx2phi} |
|
938 |
Let $S \sub M$ be contained in a union (not necessarily disjoint) |
|
86 | 939 |
of $k$ metric balls of radius $r$. |
89 | 940 |
Let $\phi_1, \phi_2, \ldots$ be an increasing sequence of real numbers satisfying |
941 |
$\phi_1 \ge 2$ and $\phi_{i+1} \ge \phi_i(2\phi_i + 2) + \phi_i$. |
|
942 |
For convenience, let $\phi_0 = 0$. |
|
248 | 943 |
Assume also that $\phi_k r \le \rho(M)$, |
944 |
where $\rho(M)$ is as in Lemma \ref{xxzz11}. |
|
89 | 945 |
Then there exists a neighborhood $U$ of $S$, |
946 |
homeomorphic to a disjoint union of balls, such that |
|
86 | 947 |
\[ |
89 | 948 |
\Nbd_{\phi_{k-1} r}(S) \subeq U \subeq \Nbd_{\phi_k r}(S) . |
86 | 949 |
\] |
950 |
\end{lemma} |
|
951 |
||
952 |
\begin{proof} |
|
89 | 953 |
For $k=1$ this follows from Lemma \ref{xxzz11}. |
954 |
Assume inductively that it holds for $k-1$. |
|
86 | 955 |
Partition $S$ into $k$ disjoint subsets $S_1,\ldots,S_k$, each of radius $\le r$. |
89 | 956 |
By Lemma \ref{xxzz11}, each $\Nbd_{\phi_{k-1} r}(S_i)$ is homeomorphic to a ball. |
957 |
If these balls are disjoint, let $U$ be their union. |
|
958 |
Otherwise, assume WLOG that $S_{k-1}$ and $S_k$ are distance less than $2\phi_{k-1}r$ apart. |
|
959 |
Let $R_i = \Nbd_{\phi_{k-1} r}(S_i)$ for $i = 1,\ldots,k-2$ |
|
960 |
and $R_{k-1} = \Nbd_{\phi_{k-1} r}(S_{k-1})\cup \Nbd_{\phi_{k-1} r}(S_k)$. |
|
961 |
Each $R_i$ is contained in a metric ball of radius $r' \deq (2\phi_{k-1}+2)r$. |
|
91 | 962 |
Note that the defining inequality of the $\phi_i$ guarantees that |
963 |
\[ |
|
964 |
\phi_{k-1}r' = \phi_{k-1}(2\phi_{k-1}+2)r \le \phi_k r \le \rho(M) . |
|
965 |
\] |
|
89 | 966 |
By induction, there is a neighborhood $U$ of $R \deq \bigcup_i R_i$, |
967 |
homeomorphic to a disjoint union |
|
968 |
of balls, and such that |
|
86 | 969 |
\[ |
89 | 970 |
U \subeq \Nbd_{\phi_{k-1}r'}(R) = \Nbd_{t}(S) \subeq \Nbd_{\phi_k r}(S) , |
86 | 971 |
\] |
89 | 972 |
where $t = \phi_{k-1}(2\phi_{k-1}+2)r + \phi_{k-1} r$. |
86 | 973 |
\end{proof} |
974 |
||
430 | 975 |
|
976 |
We now return to defining the chain maps $e_X$. |
|
977 |
||
70 | 978 |
|
254 | 979 |
Let $R_*$ be the chain complex with a generating 0-chain for each non-negative |
980 |
integer and a generating 1-chain connecting each adjacent pair $(j, j+1)$. |
|
358
8589275ac65b
CH_* action -- gluing compatibility
Kevin Walker <kevin@canyon23.net>
parents:
357
diff
changeset
|
981 |
(So $R_*$ is a simplicial version of the non-negative reals.) |
254 | 982 |
Denote the 0-chains by $j$ (for $j$ a non-negative integer) and the 1-chain connecting $j$ and $j+1$ |
983 |
by $\iota_j$. |
|
984 |
Define a map (homotopy equivalence) |
|
250 | 985 |
\[ |
254 | 986 |
\sigma: R_*\ot CH_*(X, X) \otimes \bc_*(X) \to CH_*(X, X)\ot \bc_*(X) |
250 | 987 |
\] |
254 | 988 |
as follows. |
989 |
On $R_0\ot CH_*(X, X) \otimes \bc_*(X)$ we define |
|
990 |
\[ |
|
991 |
\sigma(j\ot p\ot b) = g_j(p)\ot b . |
|
992 |
\] |
|
255 | 993 |
On $R_1\ot CH_*(X, X) \otimes \bc_*(X)$ we define |
994 |
\[ |
|
995 |
\sigma(\iota_j\ot p\ot b) = f_j(p)\ot b , |
|
996 |
\] |
|
997 |
where $f_j$ is the homotopy from $g_j$ to $g_{j+1}$. |
|
86 | 998 |
|
254 | 999 |
Next we specify subcomplexes $G^m_* \sub R_*\ot CH_*(X, X) \otimes \bc_*(X)$ on which we will eventually |
1000 |
define a version of the action map $e_X$. |
|
255 | 1001 |
A generator $j\ot p\ot b$ is defined to be in $G^m_*$ if $j\ge j_{kbmp}$, where |
254 | 1002 |
$k = k_{bmp}$ is the constant from Lemma \ref{Gim_approx}. |
255 | 1003 |
Similarly $\iota_j\ot p\ot b$ is in $G^m_*$ if $j\ge j_{kbmp}$. |
254 | 1004 |
The inequality following Lemma \ref{Gim_approx} guarantees that $G^m_*$ is indeed a subcomplex |
1005 |
and that $G^m_* \sup G^{m+1}_*$. |
|
250 | 1006 |
|
254 | 1007 |
It is easy to see that each $G^m_*$ is homotopy equivalent (via the inclusion map) |
1008 |
to $R_*\ot CH_*(X, X) \otimes \bc_*(X)$ |
|
1009 |
and hence to $CH_*(X, X) \otimes \bc_*(X)$, and furthermore that the homotopies are well-defined |
|
1010 |
up to a contractible set of choices. |
|
250 | 1011 |
|
254 | 1012 |
Next we define a map |
1013 |
\[ |
|
1014 |
e_m : G^m_* \to \bc_*(X) . |
|
1015 |
\] |
|
255 | 1016 |
Let $p\ot b$ be a generator of $G^m_*$. |
1017 |
Each $g_j(p)\ot b$ or $f_j(p)\ot b$ is a linear combination of generators $q\ot c$, |
|
1018 |
where $\supp(q)\cup\supp(c)$ is contained in a disjoint union of balls satisfying |
|
1019 |
various conditions specified above. |
|
1020 |
As in the construction of the maps $e_{i,m}$ above, |
|
1021 |
it suffices to specify for each such $q\ot c$ a disjoint union of balls |
|
1022 |
$V_{qc} \sup \supp(q)\cup\supp(c)$, such that $V_{qc} \sup V_{q'c'}$ |
|
1023 |
whenever $q'\ot c'$ appears in the boundary of $q\ot c$. |
|
1024 |
||
1025 |
Let $q\ot c$ be a summand of $g_j(p)\ot b$, as above. |
|
1026 |
Let $i$ be maximal such that $j\ge j_{ibmp}$ |
|
1027 |
(notation as in Lemma \ref{Gim_approx}). |
|
1028 |
Then $q\ot c \in G^{i,m}_*$ and we choose $V_{qc} \sup \supp(q)\cup\supp(c)$ |
|
1029 |
such that |
|
1030 |
\[ |
|
1031 |
N_{i,d}(q\ot c) \subeq V_{qc} \subeq N_{i,d+1}(q\ot c) , |
|
1032 |
\] |
|
1033 |
where $d = \deg(q\ot c)$. |
|
1034 |
Let $\tilde q = f_j(q)$. |
|
1035 |
The summands of $f_j(p)\ot b$ have the form $\tilde q \ot c$, |
|
1036 |
where $q\ot c$ is a summand of $g_j(p)\ot b$. |
|
1037 |
Since the homotopy $f_j$ does not increase supports, we also have that |
|
1038 |
\[ |
|
1039 |
V_{qc} \sup \supp(\tilde q) \cup \supp(c) . |
|
1040 |
\] |
|
1041 |
So we define $V_{\tilde qc} = V_{qc}$. |
|
1042 |
||
1043 |
It is now easy to check that we have $V_{qc} \sup V_{q'c'}$ |
|
1044 |
whenever $q'\ot c'$ appears in the boundary of $q\ot c$. |
|
1045 |
As in the construction of the maps $e_{i,m}$ above, |
|
1046 |
this allows us to construct a map |
|
1047 |
\[ |
|
1048 |
e_m : G^m_* \to \bc_*(X) |
|
1049 |
\] |
|
1050 |
which is well-defined up to homotopy. |
|
1051 |
As in the proof of Lemma \ref{m_order_hty}, we can show that the map is well-defined up |
|
1052 |
to $m$-th order homotopy. |
|
1053 |
Put another way, we have specified an $m$-connected subcomplex of the complex of |
|
1054 |
all maps $G^m_* \to \bc_*(X)$. |
|
1055 |
On $G^{m+1}_* \sub G^m_*$ we have defined two maps, $e_m$ and $e_{m+1}$. |
|
1056 |
One can similarly (to the proof of Lemma \ref{m_order_hty}) show that |
|
1057 |
these two maps agree up to $m$-th order homotopy. |
|
1058 |
More precisely, one can show that the subcomplex of maps containing the various |
|
1059 |
$e_{m+1}$ candidates is contained in the corresponding subcomplex for $e_m$. |
|
253
3816f6ce80a8
evmap; about to delete a few paragraphs, but committing just so there's
Kevin Walker <kevin@canyon23.net>
parents:
251
diff
changeset
|
1060 |
|
358
8589275ac65b
CH_* action -- gluing compatibility
Kevin Walker <kevin@canyon23.net>
parents:
357
diff
changeset
|
1061 |
\medskip |
8589275ac65b
CH_* action -- gluing compatibility
Kevin Walker <kevin@canyon23.net>
parents:
357
diff
changeset
|
1062 |
|
8589275ac65b
CH_* action -- gluing compatibility
Kevin Walker <kevin@canyon23.net>
parents:
357
diff
changeset
|
1063 |
Next we show that the action maps are compatible with gluing. |
8589275ac65b
CH_* action -- gluing compatibility
Kevin Walker <kevin@canyon23.net>
parents:
357
diff
changeset
|
1064 |
Let $G^m_*$ and $\ol{G}^m_*$ be the complexes, as above, used for defining |
8589275ac65b
CH_* action -- gluing compatibility
Kevin Walker <kevin@canyon23.net>
parents:
357
diff
changeset
|
1065 |
the action maps $e_{X\sgl}$ and $e_X$. |
8589275ac65b
CH_* action -- gluing compatibility
Kevin Walker <kevin@canyon23.net>
parents:
357
diff
changeset
|
1066 |
The gluing map $X\sgl\to X$ induces a map |
8589275ac65b
CH_* action -- gluing compatibility
Kevin Walker <kevin@canyon23.net>
parents:
357
diff
changeset
|
1067 |
\[ |
430 | 1068 |
\gl: R_*\ot CH_*(X, X) \otimes \bc_*(X) \to R_*\ot CH_*(X\sgl, X \sgl) \otimes \bc_*(X \sgl) , |
358
8589275ac65b
CH_* action -- gluing compatibility
Kevin Walker <kevin@canyon23.net>
parents:
357
diff
changeset
|
1069 |
\] |
8589275ac65b
CH_* action -- gluing compatibility
Kevin Walker <kevin@canyon23.net>
parents:
357
diff
changeset
|
1070 |
and it is easy to see that $\gl(G^m_*)\sub \ol{G}^m_*$. |
437 | 1071 |
From this it follows that the diagram in the statement of Theorem \ref{thm:CH} commutes. |
358
8589275ac65b
CH_* action -- gluing compatibility
Kevin Walker <kevin@canyon23.net>
parents:
357
diff
changeset
|
1072 |
|
430 | 1073 |
\todo{this paragraph isn't very convincing, or at least I don't see what's going on} |
358
8589275ac65b
CH_* action -- gluing compatibility
Kevin Walker <kevin@canyon23.net>
parents:
357
diff
changeset
|
1074 |
Finally we show that the action maps defined above are independent of |
8589275ac65b
CH_* action -- gluing compatibility
Kevin Walker <kevin@canyon23.net>
parents:
357
diff
changeset
|
1075 |
the choice of metric (up to iterated homotopy). |
359
6224e50c9311
metric independence for homeo action (proof done now)
Kevin Walker <kevin@canyon23.net>
parents:
358
diff
changeset
|
1076 |
The arguments are very similar to ones given above, so we only sketch them. |
6224e50c9311
metric independence for homeo action (proof done now)
Kevin Walker <kevin@canyon23.net>
parents:
358
diff
changeset
|
1077 |
Let $g$ and $g'$ be two metrics on $X$, and let $e$ and $e'$ be the corresponding |
6224e50c9311
metric independence for homeo action (proof done now)
Kevin Walker <kevin@canyon23.net>
parents:
358
diff
changeset
|
1078 |
actions $CH_*(X, X) \ot \bc_*(X)\to\bc_*(X)$. |
6224e50c9311
metric independence for homeo action (proof done now)
Kevin Walker <kevin@canyon23.net>
parents:
358
diff
changeset
|
1079 |
We must show that $e$ and $e'$ are homotopic. |
6224e50c9311
metric independence for homeo action (proof done now)
Kevin Walker <kevin@canyon23.net>
parents:
358
diff
changeset
|
1080 |
As outlined in the discussion preceding this proof, |
6224e50c9311
metric independence for homeo action (proof done now)
Kevin Walker <kevin@canyon23.net>
parents:
358
diff
changeset
|
1081 |
this follows from the facts that both $e$ and $e'$ are compatible |
6224e50c9311
metric independence for homeo action (proof done now)
Kevin Walker <kevin@canyon23.net>
parents:
358
diff
changeset
|
1082 |
with gluing and that $\bc_*(B^n)$ is contractible. |
6224e50c9311
metric independence for homeo action (proof done now)
Kevin Walker <kevin@canyon23.net>
parents:
358
diff
changeset
|
1083 |
As above, we define a subcomplex $F_*\sub CH_*(X, X) \ot \bc_*(X)$ generated |
6224e50c9311
metric independence for homeo action (proof done now)
Kevin Walker <kevin@canyon23.net>
parents:
358
diff
changeset
|
1084 |
by $p\ot b$ such that $|p|\cup|b|$ is contained in a disjoint union of balls. |
6224e50c9311
metric independence for homeo action (proof done now)
Kevin Walker <kevin@canyon23.net>
parents:
358
diff
changeset
|
1085 |
Using acyclic models, we can construct a homotopy from $e$ to $e'$ on $F_*$. |
6224e50c9311
metric independence for homeo action (proof done now)
Kevin Walker <kevin@canyon23.net>
parents:
358
diff
changeset
|
1086 |
We now observe that $CH_*(X, X) \ot \bc_*(X)$ retracts to $F_*$. |
6224e50c9311
metric independence for homeo action (proof done now)
Kevin Walker <kevin@canyon23.net>
parents:
358
diff
changeset
|
1087 |
Similar arguments show that this homotopy from $e$ to $e'$ is well-defined |
6224e50c9311
metric independence for homeo action (proof done now)
Kevin Walker <kevin@canyon23.net>
parents:
358
diff
changeset
|
1088 |
up to second order homotopy, and so on. |
430 | 1089 |
|
437 | 1090 |
This completes the proof of Theorem \ref{thm:CH}. |
84 | 1091 |
\end{proof} |
1092 |
||
1093 |
||
396
f58d590e8a08
cross-references for the small blobs lemma
Scott Morrison <scott@tqft.net>
parents:
385
diff
changeset
|
1094 |
\begin{rem*} |
f58d590e8a08
cross-references for the small blobs lemma
Scott Morrison <scott@tqft.net>
parents:
385
diff
changeset
|
1095 |
\label{rem:for-small-blobs} |
f58d590e8a08
cross-references for the small blobs lemma
Scott Morrison <scott@tqft.net>
parents:
385
diff
changeset
|
1096 |
For the proof of Lemma \ref{lem:CH-small-blobs} below we will need the following observation on the action constructed above. |
368
eb7a1ea85179
aborted attempt at remark for small blobs lemma
Kevin Walker <kevin@canyon23.net>
parents:
359
diff
changeset
|
1097 |
Let $b$ be a blob diagram and $p:P\times X\to X$ be a family of homeomorphisms. |
eb7a1ea85179
aborted attempt at remark for small blobs lemma
Kevin Walker <kevin@canyon23.net>
parents:
359
diff
changeset
|
1098 |
Then we may choose $e$ such that $e(p\ot b)$ is a sum of generators, each |
385
b1da2a454ee7
refinement of ev map statement needed for small blobs
Kevin Walker <kevin@canyon23.net>
parents:
368
diff
changeset
|
1099 |
of which has support close to $p(t,|b|)$ for some $t\in P$. |
430 | 1100 |
More precisely, the support of the generators is contained in the union of a small neighborhood |
1101 |
of $p(t,|b|)$ with some small balls. |
|
385
b1da2a454ee7
refinement of ev map statement needed for small blobs
Kevin Walker <kevin@canyon23.net>
parents:
368
diff
changeset
|
1102 |
(Here ``small" is in terms of the metric on $X$ that we chose to construct $e$.) |
396
f58d590e8a08
cross-references for the small blobs lemma
Scott Morrison <scott@tqft.net>
parents:
385
diff
changeset
|
1103 |
\end{rem*} |
385
b1da2a454ee7
refinement of ev map statement needed for small blobs
Kevin Walker <kevin@canyon23.net>
parents:
368
diff
changeset
|
1104 |
|
b1da2a454ee7
refinement of ev map statement needed for small blobs
Kevin Walker <kevin@canyon23.net>
parents:
368
diff
changeset
|
1105 |
|
437 | 1106 |
\begin{thm} |
1107 |
\label{thm:CH-associativity} |
|
357
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1108 |
The $CH_*(X, Y)$ actions defined above are associative. |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1109 |
That is, the following diagram commutes up to homotopy: |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1110 |
\[ \xymatrix{ |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1111 |
& CH_*(Y, Z) \ot \bc_*(Y) \ar[dr]^{e_{YZ}} & \\ |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1112 |
CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & \bc_*(Z) \\ |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1113 |
& CH_*(X, Z) \ot \bc_*(X) \ar[ur]_{e_{XZ}} & |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1114 |
} \] |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1115 |
Here $\mu:CH_*(X, Y) \ot CH_*(Y, Z)\to CH_*(X, Z)$ is the map induced by composition |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1116 |
of homeomorphisms. |
437 | 1117 |
\end{thm} |
70 | 1118 |
|
357
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1119 |
\begin{proof} |
437 | 1120 |
The strategy of the proof is similar to that of Theorem \ref{thm:CH}. |
357
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1121 |
We will identify a subcomplex |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1122 |
\[ |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1123 |
G_* \sub CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1124 |
\] |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1125 |
where it is easy to see that the two sides of the diagram are homotopic, then |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1126 |
show that there is a deformation retraction of $CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)$ into $G_*$. |
70 | 1127 |
|
357
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1128 |
Let $p\ot q\ot b$ be a generator of $CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)$. |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1129 |
By definition, $p\ot q\ot b\in G_*$ if there is a disjoint union of balls in $X$ which |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1130 |
contains $|p| \cup p\inv(|q|) \cup |b|$. |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1131 |
(If $p:P\times X\to Y$, then $p\inv(|q|)$ means the union over all $x\in P$ of |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1132 |
$p(x, \cdot)\inv(|q|)$.) |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1133 |
|
437 | 1134 |
As in the proof of Theorem \ref{thm:CH}, we can construct a homotopy |
357
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1135 |
between the upper and lower maps restricted to $G_*$. |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1136 |
This uses the facts that the maps agree on $CH_0(X, Y) \ot CH_0(Y, Z) \ot \bc_*(X)$, |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1137 |
that they are compatible with gluing, and the contractibility of $\bc_*(X)$. |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1138 |
|
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1139 |
We can now apply Lemma \ref{extension_lemma_c}, using a series of increasingly fine covers, |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1140 |
to construct a deformation retraction of $CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)$ into $G_*$. |
bbd55b6e9650
associativity for CH_* action
Kevin Walker <kevin@canyon23.net>
parents:
345
diff
changeset
|
1141 |
\end{proof} |
524
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
1142 |
|
edf8798ef477
finished 1st draft of new evmap section; commented out older versions
Kevin Walker <kevin@canyon23.net>
parents:
523
diff
changeset
|
1143 |
} % end \noop |