author | Scott Morrison <scott@tqft.net> |
Sun, 20 Jun 2010 11:00:57 -0700 | |
changeset 378 | e5867a64cae5 |
parent 376 | 1ce74e992a72 |
child 381 | 84bcc5fdf8c2 |
permissions | -rw-r--r-- |
149 | 1 |
%!TEX root = ../blob1.tex |
2 |
||
3 |
\section{Higher-dimensional Deligne conjecture} |
|
4 |
\label{sec:deligne} |
|
288 | 5 |
In this section we |
6 |
sketch |
|
7 |
\nn{revisit ``sketch" after proof is done} |
|
8 |
the proof of a higher dimensional version of the Deligne conjecture |
|
9 |
about the action of the little disks operad on Hochschild cohomology. |
|
10 |
The first several paragraphs lead up to a precise statement of the result |
|
11 |
(Proposition \ref{prop:deligne} below). |
|
12 |
Then we sketch the proof. |
|
163 | 13 |
|
342 | 14 |
\nn{Does this generalisation encompass Kontsevich's proposed generalisation from \cite[\S2.5]{MR1718044}, |
15 |
that (I think...) the Hochschild homology of an $E_n$ algebra is an $E_{n+1}$ algebra? -S} |
|
292
7d0c63a9ce05
adding some biblio entries re: Deligne. Run svn up bibliography to update the bibliography, which is still in SVN
Scott Morrison <scott@tqft.net>
parents:
289
diff
changeset
|
16 |
|
7d0c63a9ce05
adding some biblio entries re: Deligne. Run svn up bibliography to update the bibliography, which is still in SVN
Scott Morrison <scott@tqft.net>
parents:
289
diff
changeset
|
17 |
%from http://www.ams.org/mathscinet-getitem?mr=1805894 |
7d0c63a9ce05
adding some biblio entries re: Deligne. Run svn up bibliography to update the bibliography, which is still in SVN
Scott Morrison <scott@tqft.net>
parents:
289
diff
changeset
|
18 |
%Different versions of the geometric counterpart of Deligne's conjecture have been proven by Tamarkin [``Formality of chain operad of small squares'', preprint, http://arXiv.org/abs/math.QA/9809164], the reviewer [in Confˇrence Moshˇ Flato 1999, Vol. II (Dijon), 307--331, Kluwer Acad. Publ., Dordrecht, 2000; MR1805923 (2002d:55009)], and J. E. McClure and J. H. Smith [``A solution of Deligne's conjecture'', preprint, http://arXiv.org/abs/math.QA/9910126] (see also a later simplified version [J. E. McClure and J. H. Smith, ``Multivariable cochain operations and little $n$-cubes'', preprint, http://arXiv.org/abs/math.QA/0106024]). The paper under review gives another proof of Deligne's conjecture, which, as the authors indicate, may be generalized to a proof of a higher-dimensional generalization of Deligne's conjecture, suggested in [M. Kontsevich, Lett. Math. Phys. 48 (1999), no. 1, 35--72; MR1718044 (2000j:53119)]. |
7d0c63a9ce05
adding some biblio entries re: Deligne. Run svn up bibliography to update the bibliography, which is still in SVN
Scott Morrison <scott@tqft.net>
parents:
289
diff
changeset
|
19 |
|
7d0c63a9ce05
adding some biblio entries re: Deligne. Run svn up bibliography to update the bibliography, which is still in SVN
Scott Morrison <scott@tqft.net>
parents:
289
diff
changeset
|
20 |
|
7d0c63a9ce05
adding some biblio entries re: Deligne. Run svn up bibliography to update the bibliography, which is still in SVN
Scott Morrison <scott@tqft.net>
parents:
289
diff
changeset
|
21 |
The usual Deligne conjecture (proved variously in \cite{MR1805894, MR2064592, hep-th/9403055, MR1805923} gives a map |
163 | 22 |
\[ |
23 |
C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
|
24 |
\to Hoch^*(C, C) . |
|
25 |
\] |
|
26 |
Here $LD_k$ is the $k$-th space of the little disks operad, and $Hoch^*(C, C)$ denotes Hochschild |
|
27 |
cochains. |
|
288 | 28 |
The little disks operad is homotopy equivalent to the |
29 |
(transversely orient) fat graph operad |
|
30 |
\nn{need ref, or say more precisely what we mean}, |
|
31 |
and Hochschild cochains are homotopy equivalent to $A_\infty$ endomorphisms |
|
32 |
of the blob complex of the interval, thought of as a bimodule for itself. |
|
163 | 33 |
\nn{need to make sure we prove this above}. |
34 |
So the 1-dimensional Deligne conjecture can be restated as |
|
283
418919afd077
small preliminary changes to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
237
diff
changeset
|
35 |
\[ |
418919afd077
small preliminary changes to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
237
diff
changeset
|
36 |
C_*(FG_k)\otimes \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots |
418919afd077
small preliminary changes to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
237
diff
changeset
|
37 |
\otimes \hom(\bc^C_*(I), \bc^C_*(I)) |
418919afd077
small preliminary changes to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
237
diff
changeset
|
38 |
\to \hom(\bc^C_*(I), \bc^C_*(I)) . |
418919afd077
small preliminary changes to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
237
diff
changeset
|
39 |
\] |
163 | 40 |
See Figure \ref{delfig1}. |
301
f956f235213a
adding some figures to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
300
diff
changeset
|
41 |
\begin{figure}[t] |
237
d42ae7a54143
diagrams for deligne conjecture, and more work on small blobs
Scott Morrison <scott@tqft.net>
parents:
194
diff
changeset
|
42 |
$$\mathfig{.9}{deligne/intervals}$$ |
163 | 43 |
\caption{A fat graph}\label{delfig1}\end{figure} |
288 | 44 |
We emphasize that in $\hom(\bc^C_*(I), \bc^C_*(I))$ we are thinking of $\bc^C_*(I)$ as a module |
45 |
for the $A_\infty$ 1-category associated to $\bd I$, and $\hom$ means the |
|
46 |
morphisms of such modules as defined in |
|
47 |
Subsection \ref{ss:module-morphisms}. |
|
163 | 48 |
|
49 |
We can think of a fat graph as encoding a sequence of surgeries, starting at the bottommost interval |
|
50 |
of Figure \ref{delfig1} and ending at the topmost interval. |
|
51 |
The surgeries correspond to the $k$ bigon-shaped ``holes" in the fat graph. |
|
52 |
We remove the bottom interval of the bigon and replace it with the top interval. |
|
288 | 53 |
To convert this topological operation to an algebraic one, we need, for each hole, an element of |
283
418919afd077
small preliminary changes to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
237
diff
changeset
|
54 |
$\hom(\bc^C_*(I_{\text{bottom}}), \bc^C_*(I_{\text{top}}))$. |
163 | 55 |
So for each fixed fat graph we have a map |
56 |
\[ |
|
283
418919afd077
small preliminary changes to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
237
diff
changeset
|
57 |
\hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots |
418919afd077
small preliminary changes to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
237
diff
changeset
|
58 |
\otimes \hom(\bc^C_*(I), \bc^C_*(I)) \to \hom(\bc^C_*(I), \bc^C_*(I)) . |
163 | 59 |
\] |
60 |
If we deform the fat graph, corresponding to a 1-chain in $C_*(FG_k)$, we get a homotopy |
|
61 |
between the maps associated to the endpoints of the 1-chain. |
|
62 |
Similarly, higher-dimensional chains in $C_*(FG_k)$ give rise to higher homotopies. |
|
63 |
||
64 |
It should now be clear how to generalize this to higher dimensions. |
|
65 |
In the sequence-of-surgeries description above, we never used the fact that the manifolds |
|
66 |
involved were 1-dimensional. |
|
288 | 67 |
Thus we can define an $n$-dimensional fat graph to be a sequence of general surgeries |
289
7c26ae009b75
adding more detail to def of n-dim fat graph operad
Kevin Walker <kevin@canyon23.net>
parents:
288
diff
changeset
|
68 |
on an $n$-manifold (Figure \ref{delfig2}). |
301
f956f235213a
adding some figures to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
300
diff
changeset
|
69 |
\begin{figure}[t] |
237
d42ae7a54143
diagrams for deligne conjecture, and more work on small blobs
Scott Morrison <scott@tqft.net>
parents:
194
diff
changeset
|
70 |
$$\mathfig{.9}{deligne/manifolds}$$ |
301
f956f235213a
adding some figures to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
300
diff
changeset
|
71 |
\caption{An $n$-dimensional fat graph}\label{delfig2} |
288 | 72 |
\end{figure} |
73 |
||
300
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
74 |
More specifically, an $n$-dimensional fat graph ($n$-FG for short) consists of: |
289
7c26ae009b75
adding more detail to def of n-dim fat graph operad
Kevin Walker <kevin@canyon23.net>
parents:
288
diff
changeset
|
75 |
\begin{itemize} |
298 | 76 |
\item ``Upper" $n$-manifolds $M_0,\ldots,M_k$ and ``lower" $n$-manifolds $N_0,\ldots,N_k$, |
77 |
with $\bd M_i = \bd N_i = E_i$ for all $i$. |
|
78 |
We call $M_0$ and $N_0$ the outer boundary and the remaining $M_i$'s and $N_i$'s the inner |
|
79 |
boundaries. |
|
80 |
\item Additional manifolds $R_1,\ldots,R_{k}$, with $\bd R_i = E_0\cup \bd M_i = E_0\cup \bd N_i$. |
|
81 |
%(By convention, $M_i = N_i = \emptyset$ if $i <1$ or $i>k$.) |
|
82 |
\item Homeomorphisms |
|
83 |
\begin{eqnarray*} |
|
84 |
f_0: M_0 &\to& R_1\cup M_1 \\ |
|
85 |
f_i: R_i\cup N_i &\to& R_{i+1}\cup M_{i+1}\;\; \mbox{for}\, 1\le i \le k-1 \\ |
|
86 |
f_k: R_k\cup N_k &\to& N_0 . |
|
87 |
\end{eqnarray*} |
|
88 |
Each $f_i$ should be the identity restricted to $E_0$. |
|
289
7c26ae009b75
adding more detail to def of n-dim fat graph operad
Kevin Walker <kevin@canyon23.net>
parents:
288
diff
changeset
|
89 |
\end{itemize} |
7c26ae009b75
adding more detail to def of n-dim fat graph operad
Kevin Walker <kevin@canyon23.net>
parents:
288
diff
changeset
|
90 |
We can think of the above data as encoding the union of the mapping cylinders $C(f_0),\ldots,C(f_k)$, |
295 | 91 |
with $C(f_i)$ glued to $C(f_{i+1})$ along $R_{i+1}$ |
301
f956f235213a
adding some figures to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
300
diff
changeset
|
92 |
(see Figure \ref{xdfig2}). |
f956f235213a
adding some figures to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
300
diff
changeset
|
93 |
\begin{figure}[t] |
370
0680b6cd90cf
beginning to update deligne diagrams
Scott Morrison <scott@tqft.net>
parents:
354
diff
changeset
|
94 |
$$\mathfig{.9}{deligne/mapping-cylinders}$$ |
0680b6cd90cf
beginning to update deligne diagrams
Scott Morrison <scott@tqft.net>
parents:
354
diff
changeset
|
95 |
\caption{An $n$-dimensional fat graph constructed from mapping cylinders}\label{xdfig2} |
301
f956f235213a
adding some figures to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
300
diff
changeset
|
96 |
\end{figure} |
295 | 97 |
The $n$-manifolds are the ``$n$-dimensional graph" and the $I$ direction of the mapping cylinders is the ``fat" part. |
98 |
We regard two such fat graphs as the same if there is a homeomorphism between them which is the |
|
99 |
identity on the boundary and which preserves the 1-dimensional fibers coming from the mapping |
|
100 |
cylinders. |
|
101 |
More specifically, we impose the following two equivalence relations: |
|
102 |
\begin{itemize} |
|
300
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
103 |
\item If $g: R_i\to R'_i$ is a homeomorphism, we can replace |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
104 |
\begin{eqnarray*} |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
105 |
(\ldots, R_{i-1}, R_i, R_{i+1}, \ldots) &\to& (\ldots, R_{i-1}, R'_i, R_{i+1}, \ldots) \\ |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
106 |
(\ldots, f_{i-1}, f_i, \ldots) &\to& (\ldots, g\circ f_{i-1}, f_i\circ g^{-1}, \ldots), |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
107 |
\end{eqnarray*} |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
108 |
leaving the $M_i$ and $N_i$ fixed. |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
109 |
(Keep in mind the case $R'_i = R_i$.) |
301
f956f235213a
adding some figures to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
300
diff
changeset
|
110 |
(See Figure \ref{xdfig3}.) |
f956f235213a
adding some figures to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
300
diff
changeset
|
111 |
\begin{figure}[t] |
376 | 112 |
$$\mathfig{.4}{deligne/dfig3a} \to \mathfig{.4}{deligne/dfig3b} $$ |
301
f956f235213a
adding some figures to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
300
diff
changeset
|
113 |
\caption{Conjugating by a homeomorphism}\label{xdfig3} |
f956f235213a
adding some figures to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
300
diff
changeset
|
114 |
\end{figure} |
295 | 115 |
\item If $M_i = M'_i \du M''_i$ and $N_i = N'_i \du N''_i$ (and there is a |
116 |
compatible disjoint union of $\bd M = \bd N$), we can replace |
|
117 |
\begin{eqnarray*} |
|
118 |
(\ldots, M_{i-1}, M_i, M_{i+1}, \ldots) &\to& (\ldots, M_{i-1}, M'_i, M''_i, M_{i+1}, \ldots) \\ |
|
119 |
(\ldots, N_{i-1}, N_i, N_{i+1}, \ldots) &\to& (\ldots, N_{i-1}, N'_i, N''_i, N_{i+1}, \ldots) \\ |
|
120 |
(\ldots, R_{i-1}, R_i, R_{i+1}, \ldots) &\to& |
|
121 |
(\ldots, R_{i-1}, R_i\cup M''_i, R_i\cup N'_i, R_{i+1}, \ldots) \\ |
|
122 |
(\ldots, f_{i-1}, f_i, \ldots) &\to& (\ldots, f_{i-1}, \rm{id}, f_i, \ldots) . |
|
123 |
\end{eqnarray*} |
|
301
f956f235213a
adding some figures to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
300
diff
changeset
|
124 |
(See Figure \ref{xdfig1}.) |
f956f235213a
adding some figures to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
300
diff
changeset
|
125 |
\begin{figure}[t] |
373
e15ad788718d
another deligne figure done
Scott Morrison <scott@tqft.net>
parents:
370
diff
changeset
|
126 |
$$\mathfig{.3}{deligne/dfig1a} \leftarrow \mathfig{.3}{deligne/dfig1b} \rightarrow \mathfig{.3}{deligne/dfig1c}$$ |
301
f956f235213a
adding some figures to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
300
diff
changeset
|
127 |
\caption{Changing the order of a surgery}\label{xdfig1} |
f956f235213a
adding some figures to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
300
diff
changeset
|
128 |
\end{figure} |
295 | 129 |
\end{itemize} |
289
7c26ae009b75
adding more detail to def of n-dim fat graph operad
Kevin Walker <kevin@canyon23.net>
parents:
288
diff
changeset
|
130 |
|
295 | 131 |
Note that the second equivalence increases the number of holes (or arity) by 1. |
300
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
132 |
We can make a similar identification with the roles of $M'_i$ and $M''_i$ reversed. |
295 | 133 |
In terms of the ``sequence of surgeries" picture, this says that if two successive surgeries |
134 |
do not overlap, we can perform them in reverse order or simultaneously. |
|
288 | 135 |
|
298 | 136 |
There is an operad structure on $n$-dimensional fat graphs, given by gluing the outer boundary |
137 |
of one graph into one of the inner boundaries of another graph. |
|
138 |
We leave it to the reader to work out a more precise statement in terms of $M_i$'s, $f_i$'s etc. |
|
139 |
||
140 |
For fixed $\ol{M} = (M_0,\ldots,M_k)$ and $\ol{N} = (N_0,\ldots,N_k)$, we let |
|
141 |
$FG^n_{\ol{M}\ol{N}}$ denote the topological space of all $n$-dimensional fat graphs as above. |
|
300
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
142 |
(Note that in different parts of $FG^n_{\ol{M}\ol{N}}$ the $M_i$'s and $N_i$'s |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
143 |
are ordered differently.) |
298 | 144 |
The topology comes from the spaces |
145 |
\[ |
|
146 |
\Homeo(M_0\to R_1\cup M_1)\times \Homeo(R_1\cup N_1\to R_2\cup M_2)\times |
|
147 |
\cdots\times \Homeo(R_k\cup N_k\to N_0) |
|
148 |
\] |
|
149 |
and the above equivalence relations. |
|
150 |
We will denote the typical element of $FG^n_{\ol{M}\ol{N}}$ by $\ol{f} = (f_0,\ldots,f_k)$. |
|
151 |
||
300
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
152 |
\medskip |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
153 |
|
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
154 |
%The little $n{+}1$-ball operad injects into the $n$-FG operad. |
349 | 155 |
The $n$-FG operad contains the little $n{+}1$-balls operad. |
300
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
156 |
Roughly speaking, given a configuration of $k$ little $n{+}1$-balls in the standard |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
157 |
$n{+}1$-ball, we fiber the complement of the balls by vertical intervals |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
158 |
and let $M_i$ [$N_i$] be the southern [northern] hemisphere of the $i$-th ball. |
349 | 159 |
More precisely, let $x_1,\ldots,x_{n+1}$ be the coordinates of $\r^{n+1}$. |
300
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
160 |
Let $z$ be a point of the $k$-th space of the little $n{+}1$-ball operad, with |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
161 |
little balls $D_1,\ldots,D_k$ inside the standard $n{+}1$-ball. |
349 | 162 |
We assume the $D_i$'s are ordered according to the $x_{n+1}$ coordinate of their centers. |
163 |
Let $\pi:\r^{n+1}\to \r^n$ be the projection corresponding to $x_{n+1}$. |
|
300
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
164 |
Let $B\sub\r^n$ be the standard $n$-ball. |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
165 |
Let $M_i$ and $N_i$ be $B$ for all $i$. |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
166 |
Identify $\pi(D_i)$ with $B$ (a.k.a.\ $M_i$ or $N_i$) via translations and dilations (no rotations). |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
167 |
Let $R_i = B\setmin \pi(D_i)$. |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
168 |
Let $f_i = \rm{id}$ for all $i$. |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
169 |
We have now defined a map from the little $n{+}1$-ball operad to the $n$-FG operad, |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
170 |
with contractible fibers. |
349 | 171 |
(The fibers correspond to moving the $D_i$'s in the $x_{n+1}$ |
172 |
direction without changing their ordering.) |
|
300
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
173 |
\nn{issue: we've described this by varying the $R_i$'s, but above we emphasize varying the $f_i$'s. |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
174 |
does this need more explanation?} |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
175 |
|
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
176 |
Another familiar subspace of the $n$-FG operad is $\Homeo(M\to N)$, which corresponds to |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
177 |
case $k=0$ (no holes). |
298 | 178 |
|
179 |
\medskip |
|
180 |
||
181 |
Let $\ol{f} \in FG^n_{\ol{M}\ol{N}}$. |
|
182 |
Let $\hom(\bc_*(M_i), \bc_*(N_i))$ denote the morphisms from $\bc_*(M_i)$ to $\bc_*(N_i)$, |
|
183 |
as modules of the $A_\infty$ 1-category $\bc_*(E_i)$. |
|
184 |
We define a map |
|
185 |
\[ |
|
186 |
p(\ol{f}): \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |
|
187 |
\to \hom(\bc_*(M_0), \bc_*(N_0)) . |
|
188 |
\] |
|
300
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
189 |
Given $\alpha_i\in\hom(\bc_*(M_i), \bc_*(N_i))$, we define $p(\ol{f}$) to be the composition |
298 | 190 |
\[ |
191 |
\bc_*(M_0) \stackrel{f_0}{\to} \bc_*(R_1\cup M_1) |
|
192 |
\stackrel{\id\ot\alpha_1}{\to} \bc_*(R_1\cup N_1) |
|
300
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
193 |
\stackrel{f_1}{\to} \bc_*(R_2\cup M_2) \stackrel{\id\ot\alpha_2}{\to} |
298 | 194 |
\cdots \stackrel{\id\ot\alpha_k}{\to} \bc_*(R_k\cup N_k) |
195 |
\stackrel{f_k}{\to} \bc_*(N_0) |
|
196 |
\] |
|
197 |
(Recall that the maps $\id\ot\alpha_i$ were defined in \nn{need ref}.) |
|
349 | 198 |
\nn{issue: haven't we only defined $\id\ot\alpha_i$ when $\alpha_i$ is closed?} |
298 | 199 |
It is easy to check that the above definition is compatible with the equivalence relations |
200 |
and also the operad structure. |
|
300
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
201 |
We can reinterpret the above as a chain map |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
202 |
\[ |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
203 |
p: C_0(FG^n_{\ol{M}\ol{N}})\ot \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
204 |
\to \hom(\bc_*(M_0), \bc_*(N_0)) . |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
205 |
\] |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
206 |
The main result of this section is that this chain map extends to the full singular |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
207 |
chain complex $C_*(FG^n_{\ol{M}\ol{N}})$. |
288 | 208 |
|
300
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
209 |
\begin{prop} |
194 | 210 |
\label{prop:deligne} |
300
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
211 |
There is a collection of chain maps |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
212 |
\[ |
283
418919afd077
small preliminary changes to Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
237
diff
changeset
|
213 |
C_*(FG^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
300
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
214 |
\hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
215 |
\] |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
216 |
which satisfy the operad compatibility conditions. |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
217 |
On $C_0(FG^n_{\ol{M}\ol{N}})$ this agrees with the chain map $p$ defined above. |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
218 |
When $k=0$, this coincides with the $C_*(\Homeo(M_0\to N_0))$ action of Section \ref{sec:evaluation}. |
194 | 219 |
\end{prop} |
167 | 220 |
|
300
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
221 |
If, in analogy to Hochschild cochains, we define elements of $\hom(M, N)$ |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
222 |
to be ``blob cochains", we can summarize the above proposition by saying that the $n$-FG operad acts on |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
223 |
blob cochains. |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
224 |
As noted above, the $n$-FG operad contains the little $n{+}1$-ball operad, so this constitutes |
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
225 |
a higher dimensional version of the Deligne conjecture for Hochschild cochains and the little 2-disk operad. |
163 | 226 |
|
349 | 227 |
\begin{proof} |
352
38da35694123
added pinched product figs
Kevin Walker <kevin@canyon23.net>
parents:
349
diff
changeset
|
228 |
As described above, $FG^n_{\overline{M}, \overline{N}}$ is equal to the disjoint |
353
3e3ff47c5350
placeholder for proof of Deligne thing
Kevin Walker <kevin@canyon23.net>
parents:
352
diff
changeset
|
229 |
union of products of homeomorphism spaces, modulo some relations. |
3e3ff47c5350
placeholder for proof of Deligne thing
Kevin Walker <kevin@canyon23.net>
parents:
352
diff
changeset
|
230 |
By Proposition \ref{CHprop} and the Eilenberg-Zilber theorem, we have for each such product $P$ |
3e3ff47c5350
placeholder for proof of Deligne thing
Kevin Walker <kevin@canyon23.net>
parents:
352
diff
changeset
|
231 |
a chain map |
3e3ff47c5350
placeholder for proof of Deligne thing
Kevin Walker <kevin@canyon23.net>
parents:
352
diff
changeset
|
232 |
\[ |
3e3ff47c5350
placeholder for proof of Deligne thing
Kevin Walker <kevin@canyon23.net>
parents:
352
diff
changeset
|
233 |
C_*(P)\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
3e3ff47c5350
placeholder for proof of Deligne thing
Kevin Walker <kevin@canyon23.net>
parents:
352
diff
changeset
|
234 |
\hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) . |
3e3ff47c5350
placeholder for proof of Deligne thing
Kevin Walker <kevin@canyon23.net>
parents:
352
diff
changeset
|
235 |
\] |
3e3ff47c5350
placeholder for proof of Deligne thing
Kevin Walker <kevin@canyon23.net>
parents:
352
diff
changeset
|
236 |
It suffices to show that the above maps are compatible with the relations whereby |
3e3ff47c5350
placeholder for proof of Deligne thing
Kevin Walker <kevin@canyon23.net>
parents:
352
diff
changeset
|
237 |
$FG^n_{\overline{M}, \overline{N}}$ is constructed from the various $P$'s. |
354
fe9295fcf31d
should read it over at least once before committing
Kevin Walker <kevin@canyon23.net>
parents:
353
diff
changeset
|
238 |
This in turn follows easily from the fact that |
fe9295fcf31d
should read it over at least once before committing
Kevin Walker <kevin@canyon23.net>
parents:
353
diff
changeset
|
239 |
the actions of $C_*(\Homeo(\cdot\to\cdot))$ are local (compatible with gluing) and associative. |
353
3e3ff47c5350
placeholder for proof of Deligne thing
Kevin Walker <kevin@canyon23.net>
parents:
352
diff
changeset
|
240 |
|
3e3ff47c5350
placeholder for proof of Deligne thing
Kevin Walker <kevin@canyon23.net>
parents:
352
diff
changeset
|
241 |
\nn{should add some detail to above} |
349 | 242 |
\end{proof} |
163 | 243 |
|
300
febbf06c3610
Deligne: defs and statement maybe done
Kevin Walker <kevin@canyon23.net>
parents:
299
diff
changeset
|
244 |
\nn{maybe point out that even for $n=1$ there's something new here.} |