48 of the blob complex, and show that they are both homotopy equivalent to $\bc_*(X)$. |
48 of the blob complex, and show that they are both homotopy equivalent to $\bc_*(X)$. |
49 |
49 |
50 \medskip |
50 \medskip |
51 |
51 |
52 If $b$ is a blob diagram in $\bc_*(X)$, recall from \S \ref{sec:basic-properties} that the {\it support} of $b$, denoted |
52 If $b$ is a blob diagram in $\bc_*(X)$, recall from \S \ref{sec:basic-properties} that the {\it support} of $b$, denoted |
53 $\supp(b)$ or $|b|$, to be the union of the blobs of $b$. |
53 $\supp(b)$ or $|b|$, is the union of the blobs of $b$. |
54 %For a general $k$-chain $a\in \bc_k(X)$, define the support of $a$ to be the union |
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. |
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 |
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 $r\in \bc_0(X\setmin S)$. |
57 $a = a'\bullet r$, where $a'\in \bc_k(S)$ and $r\in \bc_0(X\setmin S)$. |
58 |
58 |
62 We will sometimes abuse language and talk about ``the" support of $f$, |
62 We will sometimes abuse language and talk about ``the" support of $f$, |
63 again denoted $\supp(f)$ or $|f|$, to mean some particular choice of $Y$ such that |
63 again denoted $\supp(f)$ or $|f|$, to mean some particular choice of $Y$ such that |
64 $f$ is supported on $Y$. |
64 $f$ is supported on $Y$. |
65 |
65 |
66 If $f: M \cup (Y\times I) \to M$ is a collaring homeomorphism |
66 If $f: M \cup (Y\times I) \to M$ is a collaring homeomorphism |
67 (cf. end of \S \ref{ss:syst-o-fields}), |
67 (cf.\ the 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 $x\in M\setmin S$. |
68 we say that $f$ is supported on $S\sub M$ if $f(x) = x$ for all $x\in M\setmin S$. |
69 |
69 |
70 \medskip |
70 \medskip |
71 |
71 |
72 Fix $\cU$, an open cover of $X$. |
72 Fix $\cU$, an open cover of $X$. |
73 Define the ``small blob complex" $\bc^{\cU}_*(X)$ to be the subcomplex of $\bc_*(X)$ |
73 Define the ``small blob complex" $\bc^{\cU}_*(X)$ to be the subcomplex of $\bc_*(X)$ |
74 of all blob diagrams in which every blob is contained in some open set of $\cU$, |
74 generated by blob diagrams such that every blob is contained in some open set of $\cU$, |
75 and moreover each field labeling a region cut out by the blobs is splittable |
75 and moreover each field labeling a region cut out by the blobs is splittable |
76 into fields on smaller regions, each of which is contained in some open set of $\cU$. |
76 into fields on smaller regions, each of which is contained in some open set of $\cU$. |
77 |
77 |
78 \begin{lemma}[Small blobs] \label{small-blobs-b} \label{thm:small-blobs} |
78 \begin{lemma}[Small blobs] \label{small-blobs-b} \label{thm:small-blobs} |
79 The inclusion $i: \bc^{\cU}_*(X) \into \bc_*(X)$ is a homotopy equivalence. |
79 The inclusion $i: \bc^{\cU}_*(X) \into \bc_*(X)$ is a homotopy equivalence. |
112 Roughly speaking, $s(b)$ consists of a series of 1-blob diagrams implementing a series |
112 Roughly speaking, $s(b)$ consists of a series of 1-blob diagrams implementing a series |
113 of small collar maps, plus a shrunken version of $b$. |
113 of small collar maps, plus a shrunken version of $b$. |
114 The composition of all the collar maps shrinks $B$ to a ball which is small with respect to $\cU$. |
114 The composition of all the collar maps shrinks $B$ to a ball which is small with respect to $\cU$. |
115 |
115 |
116 Let $\cV_1$ be an auxiliary open cover of $X$, subordinate to $\cU$ and |
116 Let $\cV_1$ be an auxiliary open cover of $X$, subordinate to $\cU$ and |
117 fine enough that a condition stated later in the proof is satisfied. |
117 fine enough that a condition stated later in this proof is satisfied. |
118 Let $b = (B, u, r)$, with $u = \sum a_i$ the label of $B$, and $a_i\in \bc_0(B)$. |
118 Let $b = (B, u, r)$, with $u = \sum a_i$ the label of $B$, and $a_i\in \bc_0(B)$. |
119 Choose a sequence of collar maps $\bar{f}_j:B\cup\text{collar}\to B$ satisfying conditions |
119 Choose a sequence of collar maps $\bar{f}_j:B\cup\text{collar}\to B$ satisfying conditions |
120 specified at the end of this paragraph. |
120 specified at the end of this paragraph. |
121 Let $f_j:B\to B$ be the restriction of $\bar{f}_j$ to $B$; $f_j$ maps $B$ homeomorphically to |
121 Let $f_j:B\to B$ be the restriction of $\bar{f}_j$ to $B$; $f_j$ maps $B$ homeomorphically to |
122 a slightly smaller submanifold of $B$. |
122 a slightly smaller submanifold of $B$. |
424 \begin{thm} \label{thm:CH} \label{thm:evaluation}% |
424 \begin{thm} \label{thm:CH} \label{thm:evaluation}% |
425 For $n$-manifolds $X$ and $Y$ there is a chain map |
425 For $n$-manifolds $X$ and $Y$ there is a chain map |
426 \eq{ |
426 \eq{ |
427 e_{XY} : \CH{X \to Y} \otimes \bc_*(X) \to \bc_*(Y) , |
427 e_{XY} : \CH{X \to Y} \otimes \bc_*(X) \to \bc_*(Y) , |
428 } |
428 } |
429 well-defined up to (coherent) homotopy, |
429 well-defined up to coherent homotopy, |
430 such that |
430 such that |
431 \begin{enumerate} |
431 \begin{enumerate} |
432 \item on $C_0(\Homeo(X \to Y)) \otimes \bc_*(X)$ it agrees with the obvious action of |
432 \item on $C_0(\Homeo(X \to Y)) \otimes \bc_*(X)$ it agrees with the obvious action of |
433 $\Homeo(X, Y)$ on $\bc_*(X)$ described in Property \ref{property:functoriality}, and |
433 $\Homeo(X, Y)$ on $\bc_*(X)$ described in Property \ref{property:functoriality}, and |
434 \item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
434 \item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |