changeset 8 | 15e6335ff1d4 |
parent 7 | 4ef2f77a4652 |
child 10 | fa1a8622e792 |
7:4ef2f77a4652 | 8:15e6335ff1d4 |
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1 \documentclass[11pt,leqno]{article} |
1 \documentclass[11pt,leqno]{amsart} |
2 |
2 |
3 \usepackage{amsmath,amssymb,amsthm} |
3 \newcommand{\pathtotrunk}{./} |
4 |
4 \input{text/article_preamble.tex} |
5 \usepackage[all]{xy} |
5 \input{text/top_matter.tex} |
6 |
6 |
7 % test edit #3 |
7 % test edit #3 |
8 |
8 |
9 %%%%% excerpts from my include file of standard macros |
9 %%%%% excerpts from my include file of standard macros |
10 |
10 |
11 \def\bc{{\cal B}} |
11 \def\bc{{\mathcal B}} |
12 |
12 |
13 \def\z{\mathbb{Z}} |
13 \def\z{\mathbb{Z}} |
14 \def\r{\mathbb{R}} |
14 \def\r{\mathbb{R}} |
15 \def\c{\mathbb{C}} |
15 \def\c{\mathbb{C}} |
16 \def\t{\mathbb{T}} |
16 \def\t{\mathbb{T}} |
36 \newcommand{\eqspl}[1]{\begin{displaymath}\begin{split}#1\end{split}\end{displaymath}} |
36 \newcommand{\eqspl}[1]{\begin{displaymath}\begin{split}#1\end{split}\end{displaymath}} |
37 |
37 |
38 % tricky way to iterate macros over a list |
38 % tricky way to iterate macros over a list |
39 \def\semicolon{;} |
39 \def\semicolon{;} |
40 \def\applytolist#1{ |
40 \def\applytolist#1{ |
41 \expandafter\def\csname multi#1\endcsname##1{ |
41 \expandafter\def\csname multi#1\endcsname##1{ |
42 \def\multiack{##1}\ifx\multiack\semicolon |
42 \def\multiack{##1}\ifx\multiack\semicolon |
43 \def\next{\relax} |
43 \def\next{\relax} |
44 \else |
44 \else |
45 \csname #1\endcsname{##1} |
45 \csname #1\endcsname{##1} |
46 \def\next{\csname multi#1\endcsname} |
46 \def\next{\csname multi#1\endcsname} |
47 \fi |
47 \fi |
48 \next} |
48 \next} |
49 \csname multi#1\endcsname} |
49 \csname multi#1\endcsname} |
50 |
50 |
51 % \def\cA{{\cal A}} for A..Z |
51 % \def\cA{{\cal A}} for A..Z |
52 \def\calc#1{\expandafter\def\csname c#1\endcsname{{\cal #1}}} |
52 \def\calc#1{\expandafter\def\csname c#1\endcsname{{\mathcal #1}}} |
53 \applytolist{calc}QWERTYUIOPLKJHGFDSAZXCVBNM; |
53 \applytolist{calc}QWERTYUIOPLKJHGFDSAZXCVBNM; |
54 |
54 |
55 % \DeclareMathOperator{\pr}{pr} etc. |
55 % \DeclareMathOperator{\pr}{pr} etc. |
56 \def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}} |
56 \def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}} |
57 \applytolist{declaremathop}{pr}{im}{id}{gl}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{End}{Hom}{Mat}{Tet}{cat}{Diff}{sign}; |
57 \applytolist{declaremathop}{pr}{im}{id}{gl}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Diff}{sign}; |
58 |
58 |
59 |
59 |
60 |
60 |
61 %%%%%% end excerpt |
61 %%%%%% end excerpt |
62 |
62 |
72 |
72 |
73 \makeatletter |
73 \makeatletter |
74 \@addtoreset{equation}{section} |
74 \@addtoreset{equation}{section} |
75 \gdef\theequation{\thesection.\arabic{equation}} |
75 \gdef\theequation{\thesection.\arabic{equation}} |
76 \makeatother |
76 \makeatother |
77 \newtheorem{thm}[equation]{Theorem} |
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78 \newtheorem{prop}[equation]{Proposition} |
|
79 \newtheorem{lemma}[equation]{Lemma} |
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80 \newtheorem{cor}[equation]{Corollary} |
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81 \newtheorem{defn}[equation]{Definition} |
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82 |
|
83 |
77 |
84 |
78 |
85 \maketitle |
79 \maketitle |
86 |
80 |
87 \section{Introduction} |
81 \section{Introduction} |
88 |
82 |
89 (motivation, summary/outline, etc.) |
83 (motivation, summary/outline, etc.) |
90 |
84 |
91 (motivation: |
85 (motivation: |
92 (1) restore exactness in pictures-mod-relations; |
86 (1) restore exactness in pictures-mod-relations; |
93 (1') add relations-amongst-relations etc. to pictures-mod-relations; |
87 (1') add relations-amongst-relations etc. to pictures-mod-relations; |
94 (2) want answer independent of handle decomp (i.e. don't |
88 (2) want answer independent of handle decomp (i.e. don't |
95 just go from coend to derived coend (e.g. Hochschild homology)); |
89 just go from coend to derived coend (e.g. Hochschild homology)); |
96 (3) ... |
90 (3) ... |
97 ) |
91 ) |
98 |
92 |
99 \section{Definitions} |
93 \section{Definitions} |
100 |
94 |
101 \subsection{Fields} |
95 \subsection{Fields} |
102 |
96 |
103 Fix a top dimension $n$. |
97 Fix a top dimension $n$. |
104 |
98 |
105 A {\it system of fields} |
99 A {\it system of fields} |
106 \nn{maybe should look for better name; but this is the name I use elsewhere} |
100 \nn{maybe should look for better name; but this is the name I use elsewhere} |
107 is a collection of functors $\cC$ from manifolds of dimension $n$ or less |
101 is a collection of functors $\cC$ from manifolds of dimension $n$ or less |
108 to sets. |
102 to sets. |
109 These functors must satisfy various properties (see KW TQFT notes for details). |
103 These functors must satisfy various properties (see KW TQFT notes for details). |
110 For example: |
104 For example: |
111 there is a canonical identification $\cC(X \du Y) = \cC(X) \times \cC(Y)$; |
105 there is a canonical identification $\cC(X \du Y) = \cC(X) \times \cC(Y)$; |
112 there is a restriction map $\cC(X) \to \cC(\bd X)$; |
106 there is a restriction map $\cC(X) \to \cC(\bd X)$; |
113 gluing manifolds corresponds to fibered products of fields; |
107 gluing manifolds corresponds to fibered products of fields; |
114 given a field $c \in \cC(Y)$ there is a ``product field" |
108 given a field $c \in \cC(Y)$ there is a ``product field" |
115 $c\times I \in \cC(Y\times I)$; ... |
109 $c\times I \in \cC(Y\times I)$; ... |
116 \nn{should eventually include full details of definition of fields.} |
110 \nn{should eventually include full details of definition of fields.} |
117 |
111 |
118 \nn{note: probably will suppress from notation the distinction |
112 \nn{note: probably will suppress from notation the distinction |
119 between fields and their (orientation-reversal) duals} |
113 between fields and their (orientation-reversal) duals} |
120 |
114 |
121 \nn{remark that if top dimensional fields are not already linear |
115 \nn{remark that if top dimensional fields are not already linear |
122 then we will soon linearize them(?)} |
116 then we will soon linearize them(?)} |
123 |
117 |
124 The definition of a system of fields is intended to generalize |
118 The definition of a system of fields is intended to generalize |
125 the relevant properties of the following two examples of fields. |
119 the relevant properties of the following two examples of fields. |
126 |
120 |
127 The first example: Fix a target space $B$ and define $\cC(X)$ (where $X$ |
121 The first example: Fix a target space $B$ and define $\cC(X)$ (where $X$ |
128 is a manifold of dimension $n$ or less) to be the set of |
122 is a manifold of dimension $n$ or less) to be the set of |
129 all maps from $X$ to $B$. |
123 all maps from $X$ to $B$. |
130 |
124 |
131 The second example will take longer to explain. |
125 The second example will take longer to explain. |
132 Given an $n$-category $C$ with the right sort of duality |
126 Given an $n$-category $C$ with the right sort of duality |
133 (e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), |
127 (e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), |
134 we can construct a system of fields as follows. |
128 we can construct a system of fields as follows. |
135 Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
129 Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
136 with codimension $i$ cells labeled by $i$-morphisms of $C$. |
130 with codimension $i$ cells labeled by $i$-morphisms of $C$. |
137 We'll spell this out for $n=1,2$ and then describe the general case. |
131 We'll spell this out for $n=1,2$ and then describe the general case. |
138 |
132 |
147 |
141 |
148 For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
142 For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
149 an object (0-morphism) of the 1-category $C$. |
143 an object (0-morphism) of the 1-category $C$. |
150 A field on a 1-manifold $S$ consists of |
144 A field on a 1-manifold $S$ consists of |
151 \begin{itemize} |
145 \begin{itemize} |
152 \item A cell decomposition of $S$ (equivalently, a finite collection |
146 \item A cell decomposition of $S$ (equivalently, a finite collection |
153 of points in the interior of $S$); |
147 of points in the interior of $S$); |
154 \item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$) |
148 \item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$) |
155 by an object (0-morphism) of $C$; |
149 by an object (0-morphism) of $C$; |
156 \item a transverse orientation of each 0-cell, thought of as a choice of |
150 \item a transverse orientation of each 0-cell, thought of as a choice of |
157 ``domain" and ``range" for the two adjacent 1-cells; and |
151 ``domain" and ``range" for the two adjacent 1-cells; and |
158 \item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with |
152 \item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with |
159 domain and range determined by the transverse orientation and the labelings of the 1-cells. |
153 domain and range determined by the transverse orientation and the labelings of the 1-cells. |
160 \end{itemize} |
154 \end{itemize} |
161 |
155 |
162 If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels |
156 If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels |
163 of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the |
157 of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the |
164 interior of $S$, each transversely oriented and each labeled by an element (1-morphism) |
158 interior of $S$, each transversely oriented and each labeled by an element (1-morphism) |
165 of the algebra. |
159 of the algebra. |
166 |
160 |
167 \medskip |
161 \medskip |
168 |
162 |
173 A field on a 0-manifold $P$ is a labeling of each point of $P$ with |
167 A field on a 0-manifold $P$ is a labeling of each point of $P$ with |
174 an object of the 2-category $C$. |
168 an object of the 2-category $C$. |
175 A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$. |
169 A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$. |
176 A field on a 2-manifold $Y$ consists of |
170 A field on a 2-manifold $Y$ consists of |
177 \begin{itemize} |
171 \begin{itemize} |
178 \item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such |
172 \item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such |
179 that each component of the complement is homeomorphic to a disk); |
173 that each component of the complement is homeomorphic to a disk); |
180 \item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$) |
174 \item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$) |
181 by a 0-morphism of $C$; |
175 by a 0-morphism of $C$; |
182 \item a transverse orientation of each 1-cell, thought of as a choice of |
176 \item a transverse orientation of each 1-cell, thought of as a choice of |
183 ``domain" and ``range" for the two adjacent 2-cells; |
177 ``domain" and ``range" for the two adjacent 2-cells; |
184 \item a labeling of each 1-cell by a 1-morphism of $C$, with |
178 \item a labeling of each 1-cell by a 1-morphism of $C$, with |
185 domain and range determined by the transverse orientation of the 1-cell |
179 domain and range determined by the transverse orientation of the 1-cell |
186 and the labelings of the 2-cells; |
180 and the labelings of the 2-cells; |
187 \item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood |
181 \item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood |
188 of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped |
182 of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped |
189 to $\pm 1 \in S^1$; and |
183 to $\pm 1 \in S^1$; and |
190 \item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range |
184 \item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range |
191 determined by the labelings of the 1-cells and the parameterizations of the previous |
185 determined by the labelings of the 1-cells and the parameterizations of the previous |
192 bullet. |
186 bullet. |
193 \end{itemize} |
187 \end{itemize} |
194 \nn{need to say this better; don't try to fit everything into the bulleted list} |
188 \nn{need to say this better; don't try to fit everything into the bulleted list} |
195 |
189 |
196 For general $n$, a field on a $k$-manifold $X^k$ consists of |
190 For general $n$, a field on a $k$-manifold $X^k$ consists of |
197 \begin{itemize} |
191 \begin{itemize} |
198 \item A cell decomposition of $X$; |
192 \item A cell decomposition of $X$; |
199 \item an explicit general position homeomorphism from the link of each $j$-cell |
193 \item an explicit general position homeomorphism from the link of each $j$-cell |
200 to the boundary of the standard $(k-j)$-dimensional bihedron; and |
194 to the boundary of the standard $(k-j)$-dimensional bihedron; and |
201 \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with |
195 \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with |
202 domain and range determined by the labelings of the link of $j$-cell. |
196 domain and range determined by the labelings of the link of $j$-cell. |
203 \end{itemize} |
197 \end{itemize} |
204 |
198 |
205 %\nn{next definition might need some work; I think linearity relations should |
199 %\nn{next definition might need some work; I think linearity relations should |
206 %be treated differently (segregated) from other local relations, but I'm not sure |
200 %be treated differently (segregated) from other local relations, but I'm not sure |
207 %the next definition is the best way to do it} |
201 %the next definition is the best way to do it} |
208 |
202 |
209 \medskip |
203 \medskip |
210 |
204 |
211 For top dimensional ($n$-dimensional) manifolds, we're actually interested |
205 For top dimensional ($n$-dimensional) manifolds, we're actually interested |
212 in the linearized space of fields. |
206 in the linearized space of fields. |
213 By default, define $\cC_l(X) = \c[\cC(X)]$; that is, $\cC_l(X)$ is |
207 By default, define $\cC_l(X) = \c[\cC(X)]$; that is, $\cC_l(X)$ is |
214 the vector space of finite |
208 the vector space of finite |
215 linear combinations of fields on $X$. |
209 linear combinations of fields on $X$. |
216 If $X$ has boundary, we of course fix a boundary condition: $\cC_l(X; a) = \c[\cC(X; a)]$. |
210 If $X$ has boundary, we of course fix a boundary condition: $\cC_l(X; a) = \c[\cC(X; a)]$. |
217 Thus the restriction (to boundary) maps are well defined because we never |
211 Thus the restriction (to boundary) maps are well defined because we never |
218 take linear combinations of fields with differing boundary conditions. |
212 take linear combinations of fields with differing boundary conditions. |
219 |
213 |
220 In some cases we don't linearize the default way; instead we take the |
214 In some cases we don't linearize the default way; instead we take the |
221 spaces $\cC_l(X; a)$ to be part of the data for the system of fields. |
215 spaces $\cC_l(X; a)$ to be part of the data for the system of fields. |
222 In particular, for fields based on linear $n$-category pictures we linearize as follows. |
216 In particular, for fields based on linear $n$-category pictures we linearize as follows. |
223 Define $\cC_l(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by |
217 Define $\cC_l(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by |
224 obvious relations on 0-cell labels. |
218 obvious relations on 0-cell labels. |
225 More specifically, let $L$ be a cell decomposition of $X$ |
219 More specifically, let $L$ be a cell decomposition of $X$ |
226 and let $p$ be a 0-cell of $L$. |
220 and let $p$ be a 0-cell of $L$. |
227 Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that |
221 Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that |
228 $\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. |
222 $\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. |
229 Then the subspace $K$ is generated by things of the form |
223 Then the subspace $K$ is generated by things of the form |
230 $\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader |
224 $\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader |
231 to infer the meaning of $\alpha_{\lambda c + d}$. |
225 to infer the meaning of $\alpha_{\lambda c + d}$. |
232 Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms. |
226 Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms. |
233 |
227 |
234 \nn{Maybe comment further: if there's a natural basis of morphisms, then no need; |
228 \nn{Maybe comment further: if there's a natural basis of morphisms, then no need; |
235 will do something similar below; in general, whenever a label lives in a linear |
229 will do something similar below; in general, whenever a label lives in a linear |
236 space we do something like this; ? say something about tensor |
230 space we do something like this; ? say something about tensor |
237 product of all the linear label spaces? Yes:} |
231 product of all the linear label spaces? Yes:} |
238 |
232 |
239 For top dimensional ($n$-dimensional) manifolds, we linearize as follows. |
233 For top dimensional ($n$-dimensional) manifolds, we linearize as follows. |
240 Define an ``almost-field" to be a field without labels on the 0-cells. |
234 Define an ``almost-field" to be a field without labels on the 0-cells. |
241 (Recall that 0-cells are labeled by $n$-morphisms.) |
235 (Recall that 0-cells are labeled by $n$-morphisms.) |
242 To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism |
236 To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism |
243 space determined by the labeling of the link of the 0-cell. |
237 space determined by the labeling of the link of the 0-cell. |
244 (If the 0-cell were labeled, the label would live in this space.) |
238 (If the 0-cell were labeled, the label would live in this space.) |
245 We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). |
239 We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). |
246 We now define $\cC_l(X; a)$ to be the direct sum over all almost labelings of the |
240 We now define $\cC_l(X; a)$ to be the direct sum over all almost labelings of the |
247 above tensor products. |
241 above tensor products. |
248 |
242 |
249 |
243 |
250 |
244 |
251 \subsection{Local relations} |
245 \subsection{Local relations} |
252 |
246 |
253 Let $B^n$ denote the standard $n$-ball. |
247 Let $B^n$ denote the standard $n$-ball. |
254 A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$ |
248 A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$ |
255 (for all $c \in \cC(\bd B^n)$) satisfying the following (three?) properties. |
249 (for all $c \in \cC(\bd B^n)$) satisfying the following (three?) properties. |
256 |
250 |
257 \nn{Roughly, these are (1) the local relations imply (extended) isotopy; |
251 \nn{Roughly, these are (1) the local relations imply (extended) isotopy; |
258 (2) $U(B^n; \cdot)$ is an ideal w.r.t.\ gluing; and |
252 (2) $U(B^n; \cdot)$ is an ideal w.r.t.\ gluing; and |
259 (3) this ideal is generated by ``small" generators (contained in an open cover of $B^n$). |
253 (3) this ideal is generated by ``small" generators (contained in an open cover of $B^n$). |
260 See KW TQFT notes for details. Need to transfer details to here.} |
254 See KW TQFT notes for details. Need to transfer details to here.} |
261 |
255 |
262 For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$, |
256 For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$, |
263 where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
257 where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
264 |
258 |
290 Let $X$ be an $n$-manifold. |
284 Let $X$ be an $n$-manifold. |
291 Assume a fixed system of fields. |
285 Assume a fixed system of fields. |
292 In this section we will usually suppress boundary conditions on $X$ from the notation |
286 In this section we will usually suppress boundary conditions on $X$ from the notation |
293 (e.g. write $\cC_l(X)$ instead of $\cC_l(X; c)$). |
287 (e.g. write $\cC_l(X)$ instead of $\cC_l(X; c)$). |
294 |
288 |
295 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
289 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
296 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
290 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
297 $\overline{X \setmin Y}$. |
291 $\overline{X \setmin Y}$. |
298 |
292 |
299 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case. |
293 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case. |
300 |
294 |
324 (See xxxx above.) |
318 (See xxxx above.) |
325 \nn{maybe restate this in terms of direct sums of tensor products.} |
319 \nn{maybe restate this in terms of direct sums of tensor products.} |
326 |
320 |
327 There is a map $\bd : \bc_1(X) \to \bc_0(X)$ which sends $(B, r, u)$ to $ru$, the linear |
321 There is a map $\bd : \bc_1(X) \to \bc_0(X)$ which sends $(B, r, u)$ to $ru$, the linear |
328 combination of fields on $X$ obtained by gluing $r$ to $u$. |
322 combination of fields on $X$ obtained by gluing $r$ to $u$. |
329 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
323 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
330 just erasing the blob from the picture |
324 just erasing the blob from the picture |
331 (but keeping the blob label $u$). |
325 (but keeping the blob label $u$). |
332 |
326 |
333 Note that the skein space $A(X)$ |
327 Note that the skein space $A(X)$ |
334 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
328 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
335 |
329 |
336 $\bc_2(X)$ is the space of all relations (redundancies) among the relations of $\bc_1(X)$. |
330 $\bc_2(X)$ is the space of all relations (redundancies) among the relations of $\bc_1(X)$. |
337 More specifically, $\bc_2(X)$ is the space of all finite linear combinations of |
331 More specifically, $\bc_2(X)$ is the space of all finite linear combinations of |
338 2-blob diagrams (defined below), modulo the usual linear label relations. |
332 2-blob diagrams (defined below), modulo the usual linear label relations. |
339 \nn{and also modulo blob reordering relations?} |
333 \nn{and also modulo blob reordering relations?} |
340 |
334 |
341 \nn{maybe include longer discussion to motivate the two sorts of 2-blob diagrams} |
335 \nn{maybe include longer discussion to motivate the two sorts of 2-blob diagrams} |
342 |
336 |
401 the unlabeled blob or 0-cell. |
395 the unlabeled blob or 0-cell. |
402 Let $c = \lambda a + b$. |
396 Let $c = \lambda a + b$. |
403 Let $x_a$ be the blob diagram with label $a$, and define $x_b$ and $x_c$ similarly. |
397 Let $x_a$ be the blob diagram with label $a$, and define $x_b$ and $x_c$ similarly. |
404 Then we impose the relation |
398 Then we impose the relation |
405 \eq{ |
399 \eq{ |
406 x_c = \lambda x_a + x_b . |
400 x_c = \lambda x_a + x_b . |
407 } |
401 } |
408 \nn{should do this in terms of direct sums of tensor products} |
402 \nn{should do this in terms of direct sums of tensor products} |
409 Let $x$ and $x'$ be two blob diagrams which differ only by a permutation $\pi$ |
403 Let $x$ and $x'$ be two blob diagrams which differ only by a permutation $\pi$ |
410 of their blob labelings. |
404 of their blob labelings. |
411 Then we impose the relation |
405 Then we impose the relation |
412 \eq{ |
406 \eq{ |
413 x = \sign(\pi) x' . |
407 x = \sign(\pi) x' . |
414 } |
408 } |
415 |
409 |
416 (Alert readers will have noticed that for $k=2$ our definition |
410 (Alert readers will have noticed that for $k=2$ our definition |
417 of $\bc_k(X)$ is slightly different from the previous definition |
411 of $\bc_k(X)$ is slightly different from the previous definition |
418 of $\bc_2(X)$ --- we did not impose the reordering relations. |
412 of $\bc_2(X)$ --- we did not impose the reordering relations. |
428 if removing $B_j$ creates new twig blobs. |
422 if removing $B_j$ creates new twig blobs. |
429 If $B_l$ becomes a twig after removing $B_j$, then set $u_l = r_lu_j$, |
423 If $B_l$ becomes a twig after removing $B_j$, then set $u_l = r_lu_j$, |
430 where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
424 where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
431 Finally, define |
425 Finally, define |
432 \eq{ |
426 \eq{ |
433 \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). |
427 \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). |
434 } |
428 } |
435 The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. |
429 The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. |
436 Thus we have a chain complex. |
430 Thus we have a chain complex. |
437 |
431 |
438 \nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} |
432 \nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} |
439 |
433 |
440 |
434 |
441 \nn{TO DO: |
435 \nn{TO DO: |
442 expand definition to handle DGA and $A_\infty$ versions of $n$-categories; |
436 expand definition to handle DGA and $A_\infty$ versions of $n$-categories; |
443 relations to Chas-Sullivan string stuff} |
437 relations to Chas-Sullivan string stuff} |
444 |
438 |
445 |
439 |
446 |
440 |
447 \section{Basic properties of the blob complex} |
441 \section{Basic properties of the blob complex} |
449 \begin{prop} \label{disjunion} |
443 \begin{prop} \label{disjunion} |
450 There is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. |
444 There is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. |
451 \end{prop} |
445 \end{prop} |
452 \begin{proof} |
446 \begin{proof} |
453 Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them |
447 Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them |
454 (putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a |
448 (putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a |
455 blob diagram $(b_1, b_2)$ on $X \du Y$. |
449 blob diagram $(b_1, b_2)$ on $X \du Y$. |
456 Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way. |
450 Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way. |
457 In the other direction, any blob diagram on $X\du Y$ is equal (up to sign) |
451 In the other direction, any blob diagram on $X\du Y$ is equal (up to sign) |
458 to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines |
452 to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines |
459 a pair of blob diagrams on $X$ and $Y$. |
453 a pair of blob diagrams on $X$ and $Y$. |
465 \end{proof} |
459 \end{proof} |
466 |
460 |
467 For the next proposition we will temporarily restore $n$-manifold boundary |
461 For the next proposition we will temporarily restore $n$-manifold boundary |
468 conditions to the notation. |
462 conditions to the notation. |
469 |
463 |
470 Suppose that for all $c \in \cC(\bd B^n)$ |
464 Suppose that for all $c \in \cC(\bd B^n)$ |
471 we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$ |
465 we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$ |
472 of the quotient map |
466 of the quotient map |
473 $p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$. |
467 $p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$. |
474 \nn{always the case if we're working over $\c$}. |
468 \nn{always the case if we're working over $\c$}. |
475 Then |
469 Then |
476 \begin{prop} \label{bcontract} |
470 \begin{prop} \label{bcontract} |
488 Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to |
482 Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to |
489 the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$. |
483 the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$. |
490 \nn{$x$ is a 0-blob diagram, i.e. $x \in \cC(B^n; c)$} |
484 \nn{$x$ is a 0-blob diagram, i.e. $x \in \cC(B^n; c)$} |
491 \end{proof} |
485 \end{proof} |
492 |
486 |
493 (Note that for the above proof to work, we need the linear label relations |
487 (Note that for the above proof to work, we need the linear label relations |
494 for blob labels. |
488 for blob labels. |
495 Also we need to blob reordering relations (?).) |
489 Also we need to blob reordering relations (?).) |
496 |
490 |
497 (Note also that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy |
491 (Note also that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy |
498 equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$.) |
492 equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$.) |
523 %maybe fix the boundary and consider the category of $n$-manifolds with the given boundary.} |
517 %maybe fix the boundary and consider the category of $n$-manifolds with the given boundary.} |
524 |
518 |
525 |
519 |
526 \begin{prop} |
520 \begin{prop} |
527 For fixed fields ($n$-cat), $\bc_*$ is a functor from the category |
521 For fixed fields ($n$-cat), $\bc_*$ is a functor from the category |
528 of $n$-manifolds and diffeomorphisms to the category of chain complexes and |
522 of $n$-manifolds and diffeomorphisms to the category of chain complexes and |
529 (chain map) isomorphisms. |
523 (chain map) isomorphisms. |
530 \qed |
524 \qed |
531 \end{prop} |
525 \end{prop} |
532 |
526 |
533 \nn{need to same something about boundaries and boundary conditions above. |
527 \nn{need to same something about boundaries and boundary conditions above. |
556 $X$ to get blob diagrams on $X\sgl$: |
550 $X$ to get blob diagrams on $X\sgl$: |
557 |
551 |
558 \begin{prop} |
552 \begin{prop} |
559 There is a natural chain map |
553 There is a natural chain map |
560 \eq{ |
554 \eq{ |
561 \gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl). |
555 \gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl). |
562 } |
556 } |
563 The sum is over all fields $a$ on $Y$ compatible at their |
557 The sum is over all fields $a$ on $Y$ compatible at their |
564 ($n{-}2$-dimensional) boundaries with $c$. |
558 ($n{-}2$-dimensional) boundaries with $c$. |
565 `Natural' means natural with respect to the actions of diffeomorphisms. |
559 `Natural' means natural with respect to the actions of diffeomorphisms. |
566 \qed |
560 \qed |
567 \end{prop} |
561 \end{prop} |
568 |
562 |
572 An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$ |
566 An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$ |
573 and $X\sgl = X_1 \cup_Y X_2$. |
567 and $X\sgl = X_1 \cup_Y X_2$. |
574 (Typically one of $X_1$ or $X_2$ is a disjoint union of balls.) |
568 (Typically one of $X_1$ or $X_2$ is a disjoint union of balls.) |
575 For $x_i \in \bc_*(X_i)$, we introduce the notation |
569 For $x_i \in \bc_*(X_i)$, we introduce the notation |
576 \eq{ |
570 \eq{ |
577 x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) . |
571 x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) . |
578 } |
572 } |
579 Note that we have resumed our habit of omitting boundary labels from the notation. |
573 Note that we have resumed our habit of omitting boundary labels from the notation. |
580 |
574 |
581 |
575 |
582 \bigskip |
576 \bigskip |
587 |
581 |
588 |
582 |
589 \section{$n=1$ and Hochschild homology} |
583 \section{$n=1$ and Hochschild homology} |
590 |
584 |
591 In this section we analyze the blob complex in dimension $n=1$ |
585 In this section we analyze the blob complex in dimension $n=1$ |
592 and find that for $S^1$ the homology of the blob complex is the |
586 and find that for $S^1$ the homology of the blob complex is the |
593 Hochschild homology of the category (algebroid) that we started with. |
587 Hochschild homology of the category (algebroid) that we started with. |
594 \nn{or maybe say here that the complexes are quasi-isomorphic? in general, |
588 \nn{or maybe say here that the complexes are quasi-isomorphic? in general, |
595 should perhaps put more emphasis on the complexes and less on the homology.} |
589 should perhaps put more emphasis on the complexes and less on the homology.} |
596 |
590 |
597 Notation: $HB_i(X) = H_i(\bc_*(X))$. |
591 Notation: $HB_i(X) = H_i(\bc_*(X))$. |
598 |
592 |
599 Let us first note that there is no loss of generality in assuming that our system of |
593 Let us first note that there is no loss of generality in assuming that our system of |
600 fields comes from a category. |
594 fields comes from a category. |
601 (Or maybe (???) there {\it is} a loss of generality. |
595 (Or maybe (???) there {\it is} a loss of generality. |
602 Given any system of fields, $A(I; a, b) = \cC(I; a, b)/U(I; a, b)$ can be |
596 Given any system of fields, $A(I; a, b) = \cC(I; a, b)/U(I; a, b)$ can be |
603 thought of as the morphisms of a 1-category $C$. |
597 thought of as the morphisms of a 1-category $C$. |
604 More specifically, the objects of $C$ are $\cC(pt)$, the morphisms from $a$ to $b$ |
598 More specifically, the objects of $C$ are $\cC(pt)$, the morphisms from $a$ to $b$ |
605 are $A(I; a, b)$, and composition is given by gluing. |
599 are $A(I; a, b)$, and composition is given by gluing. |
606 If we instead take our fields to be $C$-pictures, the $\cC(pt)$ does not change |
600 If we instead take our fields to be $C$-pictures, the $\cC(pt)$ does not change |
607 and neither does $A(I; a, b) = HB_0(I; a, b)$. |
601 and neither does $A(I; a, b) = HB_0(I; a, b)$. |
622 Let $C$ be a *-1-category. |
616 Let $C$ be a *-1-category. |
623 Then specializing the definitions from above to the case $n=1$ we have: |
617 Then specializing the definitions from above to the case $n=1$ we have: |
624 \begin{itemize} |
618 \begin{itemize} |
625 \item $\cC(pt) = \ob(C)$ . |
619 \item $\cC(pt) = \ob(C)$ . |
626 \item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$. |
620 \item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$. |
627 Then an element of $\cC(R; c)$ is a collection of (transversely oriented) |
621 Then an element of $\cC(R; c)$ is a collection of (transversely oriented) |
628 points in the interior |
622 points in the interior |
629 of $R$, each labeled by a morphism of $C$. |
623 of $R$, each labeled by a morphism of $C$. |
630 The intervals between the points are labeled by objects of $C$, consistent with |
624 The intervals between the points are labeled by objects of $C$, consistent with |
631 the boundary condition $c$ and the domains and ranges of the point labels. |
625 the boundary condition $c$ and the domains and ranges of the point labels. |
632 \item There is an evaluation map $e: \cC(I; a, b) \to \mor(a, b)$ given by |
626 \item There is an evaluation map $e: \cC(I; a, b) \to \mor(a, b)$ given by |
633 composing the morphism labels of the points. |
627 composing the morphism labels of the points. |
634 Note that we also need the * of *-1-category here in order to make all the morphisms point |
628 Note that we also need the * of *-1-category here in order to make all the morphisms point |
635 the same way. |
629 the same way. |
636 \item For $x \in \mor(a, b)$ let $\chi(x) \in \cC(I; a, b)$ be the field with a single |
630 \item For $x \in \mor(a, b)$ let $\chi(x) \in \cC(I; a, b)$ be the field with a single |
637 point (at some standard location) labeled by $x$. |
631 point (at some standard location) labeled by $x$. |
638 Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the |
632 Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the |
639 form $y - \chi(e(y))$. |
633 form $y - \chi(e(y))$. |
640 Thus we can, if we choose, restrict the blob twig labels to things of this form. |
634 Thus we can, if we choose, restrict the blob twig labels to things of this form. |
641 \end{itemize} |
635 \end{itemize} |
642 |
636 |
643 We want to show that $HB_*(S^1)$ is naturally isomorphic to the |
637 We want to show that $HB_*(S^1)$ is naturally isomorphic to the |
644 Hochschild homology of $C$. |
638 Hochschild homology of $C$. |
645 \nn{Or better that the complexes are homotopic |
639 \nn{Or better that the complexes are homotopic |
646 or quasi-isomorphic.} |
640 or quasi-isomorphic.} |
647 In order to prove this we will need to extend the blob complex to allow points to also |
641 In order to prove this we will need to extend the blob complex to allow points to also |
648 be labeled by elements of $C$-$C$-bimodules. |
642 be labeled by elements of $C$-$C$-bimodules. |
689 \end{itemize} |
683 \end{itemize} |
690 |
684 |
691 First we show that $F_*(C\otimes C)$ is |
685 First we show that $F_*(C\otimes C)$ is |
692 quasi-isomorphic to the 0-step complex $C$. |
686 quasi-isomorphic to the 0-step complex $C$. |
693 |
687 |
694 Let $F'_* \sub F_*(C\otimes C)$ be the subcomplex where the label of |
688 Let $F'_* \sub F_*(C\otimes C)$ be the subcomplex where the label of |
695 the point $*$ is $1 \otimes 1 \in C\otimes C$. |
689 the point $*$ is $1 \otimes 1 \in C\otimes C$. |
696 We will show that the inclusion $i: F'_* \to F_*(C\otimes C)$ is a quasi-isomorphism. |
690 We will show that the inclusion $i: F'_* \to F_*(C\otimes C)$ is a quasi-isomorphism. |
697 |
691 |
698 Fix a small $\ep > 0$. |
692 Fix a small $\ep > 0$. |
699 Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$. |
693 Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$. |
700 Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex |
694 Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex |
701 generated by blob diagrams $b$ such that $B_\ep$ is either disjoint from |
695 generated by blob diagrams $b$ such that $B_\ep$ is either disjoint from |
702 or contained in each blob of $b$, and the two boundary points of $B_\ep$ are not labeled points of $b$. |
696 or contained in each blob of $b$, and the two boundary points of $B_\ep$ are not labeled points of $b$. |
703 For a field (picture) $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$ |
697 For a field (picture) $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$ |
704 labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$. |
698 labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$. |
705 (See Figure xxxx.) |
699 (See Figure xxxx.) |
710 Let $x \in F^\ep_*$ be a blob diagram. |
704 Let $x \in F^\ep_*$ be a blob diagram. |
711 If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $B_\ep$ to |
705 If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $B_\ep$ to |
712 $x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$. |
706 $x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$. |
713 If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows. |
707 If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows. |
714 Let $y_i$ be the restriction of $z_i$ to $B_\ep$. |
708 Let $y_i$ be the restriction of $z_i$ to $B_\ep$. |
715 Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$, |
709 Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$, |
716 and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$. |
710 and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$. |
717 Define $j_\ep(x) = \sum x_i$. |
711 Define $j_\ep(x) = \sum x_i$. |
718 \nn{need to check signs coming from blob complex differential} |
712 \nn{need to check signs coming from blob complex differential} |
719 |
713 |
720 Note that if $x \in F'_* \cap F^\ep_*$ then $j_\ep(x) \in F'_*$ also. |
714 Note that if $x \in F'_* \cap F^\ep_*$ then $j_\ep(x) \in F'_*$ also. |
721 |
715 |
722 The key property of $j_\ep$ is |
716 The key property of $j_\ep$ is |
723 \eq{ |
717 \eq{ |
724 \bd j_\ep + j_\ep \bd = \id - \sigma_\ep , |
718 \bd j_\ep + j_\ep \bd = \id - \sigma_\ep , |
725 } |
719 } |
726 where $\sigma_\ep : F^\ep_* \to F^\ep_*$ is given by replacing the restriction $y$ of each field |
720 where $\sigma_\ep : F^\ep_* \to F^\ep_*$ is given by replacing the restriction $y$ of each field |
727 mentioned in $x \in F^\ep_*$ with $s_\ep(y)$. |
721 mentioned in $x \in F^\ep_*$ with $s_\ep(y)$. |
728 Note that $\sigma_\ep(x) \in F'_*$. |
722 Note that $\sigma_\ep(x) \in F'_*$. |
729 |
723 |
730 If $j_\ep$ were defined on all of $F_*(C\otimes C)$, it would show that $\sigma_\ep$ |
724 If $j_\ep$ were defined on all of $F_*(C\otimes C)$, it would show that $\sigma_\ep$ |
731 is a homotopy inverse to the inclusion $F'_* \to F_*(C\otimes C)$. |
725 is a homotopy inverse to the inclusion $F'_* \to F_*(C\otimes C)$. |
732 One strategy would be to try to stitch together various $j_\ep$ for progressively smaller |
726 One strategy would be to try to stitch together various $j_\ep$ for progressively smaller |
733 $\ep$ and show that $F'_*$ is homotopy equivalent to $F_*(C\otimes C)$. |
727 $\ep$ and show that $F'_*$ is homotopy equivalent to $F_*(C\otimes C)$. |
734 Instead, we'll be less ambitious and just show that |
728 Instead, we'll be less ambitious and just show that |
735 $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$. |
729 $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$. |
736 |
730 |
737 If $x$ is a cycle in $F_*(C\otimes C)$, then for sufficiently small $\ep$ we have |
731 If $x$ is a cycle in $F_*(C\otimes C)$, then for sufficiently small $\ep$ we have |
738 $x \in F_*^\ep$. |
732 $x \in F_*^\ep$. |
739 (This is true for any chain in $F_*(C\otimes C)$, since chains are sums of |
733 (This is true for any chain in $F_*(C\otimes C)$, since chains are sums of |
740 finitely many blob diagrams.) |
734 finitely many blob diagrams.) |
741 Then $x$ is homologous to $s_\ep(x)$, which is in $F'_*$, so the inclusion map |
735 Then $x$ is homologous to $s_\ep(x)$, which is in $F'_*$, so the inclusion map |
742 $F'_* \sub F_*(C\otimes C)$ is surjective on homology. |
736 $F'_* \sub F_*(C\otimes C)$ is surjective on homology. |
743 If $y \in F_*(C\otimes C)$ and $\bd y = x \in F'_*$, then $y \in F^\ep_*$ for some $\ep$ |
737 If $y \in F_*(C\otimes C)$ and $\bd y = x \in F'_*$, then $y \in F^\ep_*$ for some $\ep$ |
744 and |
738 and |
745 \eq{ |
739 \eq{ |
746 \bd y = \bd (\sigma_\ep(y) + j_\ep(x)) . |
740 \bd y = \bd (\sigma_\ep(y) + j_\ep(x)) . |
747 } |
741 } |
748 Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology. |
742 Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology. |
749 This completes the proof that $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$. |
743 This completes the proof that $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$. |
750 |
744 |
751 \medskip |
745 \medskip |
767 $G''_* \to G'_*$ induces an isomorphism on $H_0$. |
761 $G''_* \to G'_*$ induces an isomorphism on $H_0$. |
768 |
762 |
769 Next we construct a degree 1 map (homotopy) $h: F'_* \to F'_*$ such that |
763 Next we construct a degree 1 map (homotopy) $h: F'_* \to F'_*$ such that |
770 for all $x \in F'_*$ we have |
764 for all $x \in F'_*$ we have |
771 \eq{ |
765 \eq{ |
772 x - \bd h(x) - h(\bd x) \in F''_* . |
766 x - \bd h(x) - h(\bd x) \in F''_* . |
773 } |
767 } |
774 Since $F'_0 = F''_0$, we can take $h_0 = 0$. |
768 Since $F'_0 = F''_0$, we can take $h_0 = 0$. |
775 Let $x \in F'_1$, with single blob $B \sub S^1$. |
769 Let $x \in F'_1$, with single blob $B \sub S^1$. |
776 If $* \notin B$, then $x \in F''_1$ and we define $h_1(x) = 0$. |
770 If $* \notin B$, then $x \in F''_1$ and we define $h_1(x) = 0$. |
777 If $* \in B$, then we work in the image of $G'_*$ and $G''_*$ (with respect to $B$). |
771 If $* \in B$, then we work in the image of $G'_*$ and $G''_*$ (with respect to $B$). |
791 \nn{need to say above more clearly and settle on notation/terminology} |
785 \nn{need to say above more clearly and settle on notation/terminology} |
792 |
786 |
793 Finally, we show that $F''_*$ is contractible. |
787 Finally, we show that $F''_*$ is contractible. |
794 \nn{need to also show that $H_0$ is the right thing; easy, but I won't do it now} |
788 \nn{need to also show that $H_0$ is the right thing; easy, but I won't do it now} |
795 Let $x$ be a cycle in $F''_*$. |
789 Let $x$ be a cycle in $F''_*$. |
796 The union of the supports of the diagrams in $x$ does not contain $*$, so there exists a |
790 The union of the supports of the diagrams in $x$ does not contain $*$, so there exists a |
797 ball $B \subset S^1$ containing the union of the supports and not containing $*$. |
791 ball $B \subset S^1$ containing the union of the supports and not containing $*$. |
798 Adding $B$ as a blob to $x$ gives a contraction. |
792 Adding $B$ as a blob to $x$ gives a contraction. |
799 \nn{need to say something else in degree zero} |
793 \nn{need to say something else in degree zero} |
800 |
794 |
801 This completes the proof that $F_*(C\otimes C)$ is |
795 This completes the proof that $F_*(C\otimes C)$ is |
811 We define a quasi-inverse \nn{right term?} $s: \bc_*(S^1) \to F_*(C)$ to the inclusion as follows. |
805 We define a quasi-inverse \nn{right term?} $s: \bc_*(S^1) \to F_*(C)$ to the inclusion as follows. |
812 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
806 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
813 * is a labeled point in $y$. |
807 * is a labeled point in $y$. |
814 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
808 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
815 Let $x \in \bc_*(S^1)$. |
809 Let $x \in \bc_*(S^1)$. |
816 Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in |
810 Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in |
817 $x$ with $y$. |
811 $x$ with $y$. |
818 It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
812 It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
819 |
813 |
820 Let $G^\ep_* \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points |
814 Let $G^\ep_* \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points |
821 in a neighborhood $B_\ep$ of *, except perhaps *. |
815 in a neighborhood $B_\ep$ of *, except perhaps *. |
822 Note that for any chain $x \in \bc_*(S^1)$, $x \in G^\ep_*$ for sufficiently small $\ep$. |
816 Note that for any chain $x \in \bc_*(S^1)$, $x \in G^\ep_*$ for sufficiently small $\ep$. |
823 \nn{rest of argument goes similarly to above} |
817 \nn{rest of argument goes similarly to above} |
824 |
818 |
825 \bigskip |
819 \bigskip |
831 Does the above exactness and contractibility guarantee such a map without writing it |
825 Does the above exactness and contractibility guarantee such a map without writing it |
832 down explicitly? |
826 down explicitly? |
833 Probably it's worth writing down an explicit map even if we don't need to.} |
827 Probably it's worth writing down an explicit map even if we don't need to.} |
834 |
828 |
835 |
829 |
836 |
830 We can also describe explicitly a map from the standard Hochschild |
837 |
831 complex to the blob complex on the circle. \nn{What properties does this |
832 map have?} |
|
833 |
|
834 \begin{figure}% |
|
835 $$\mathfig{0.6}{barycentric/barycentric}$$ |
|
836 \caption{The Hochschild chain $a \tensor b \tensor c$ is sent to |
|
837 the sum of six blob $2$-chains, corresponding to a barycentric subdivision of a $2$-simplex.} |
|
838 \label{fig:Hochschild-example}% |
|
839 \end{figure} |
|
840 |
|
841 As an example, Figure \ref{fig:Hochschild-example} shows the image of the Hochschild chain $a \tensor b \tensor c$. Only the $0$-cells are shown explicitly. |
|
842 The edges marked $x, y$ and $z$ carry the $1$-chains |
|
843 \begin{align*} |
|
844 x & = \mathfig{0.1}{barycentric/ux} & u_x = \mathfig{0.1}{barycentric/ux_ca} - \mathfig{0.1}{barycentric/ux_c-a} \\ |
|
845 y & = \mathfig{0.1}{barycentric/uy} & u_y = \mathfig{0.1}{barycentric/uy_cab} - \mathfig{0.1}{barycentric/uy_ca-b} \\ |
|
846 z & = \mathfig{0.1}{barycentric/uz} & u_z = \mathfig{0.1}{barycentric/uz_c-a-b} - \mathfig{0.1}{barycentric/uz_cab} |
|
847 \end{align*} |
|
848 and the $2$-chain labelled $A$ is |
|
849 \begin{equation*} |
|
850 A = \mathfig{0.1}{barycentric/Ax}+\mathfig{0.1}{barycentric/Ay}. |
|
851 \end{equation*} |
|
852 Note that we then have |
|
853 \begin{equation*} |
|
854 \bdy A = x+y+z. |
|
855 \end{equation*} |
|
856 |
|
857 In general, the Hochschild chain $\Tensor_{i=1}^n a_i$ is sent to the sum of $n!$ blob $(n-1)$-chains, indexed by permutations, |
|
858 $$\phi\left(\Tensor_{i=1}^n a_i) = \sum_{\pi} \phi^\pi(a_1, \ldots, a_n)$$ |
|
859 with ... |
|
838 |
860 |
839 |
861 |
840 \section{Action of $C_*(\Diff(X))$} \label{diffsect} |
862 \section{Action of $C_*(\Diff(X))$} \label{diffsect} |
841 |
863 |
842 Let $CD_*(X)$ denote $C_*(\Diff(X))$, the singular chain complex of |
864 Let $CD_*(X)$ denote $C_*(\Diff(X))$, the singular chain complex of |
847 \nn{be more restrictive here? does more need to be said?} |
869 \nn{be more restrictive here? does more need to be said?} |
848 |
870 |
849 \begin{prop} \label{CDprop} |
871 \begin{prop} \label{CDprop} |
850 For each $n$-manifold $X$ there is a chain map |
872 For each $n$-manifold $X$ there is a chain map |
851 \eq{ |
873 \eq{ |
852 e_X : CD_*(X) \otimes \bc_*(X) \to \bc_*(X) . |
874 e_X : CD_*(X) \otimes \bc_*(X) \to \bc_*(X) . |
853 } |
875 } |
854 On $CD_0(X) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X)$ on $\bc_*(X)$ |
876 On $CD_0(X) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X)$ on $\bc_*(X)$ |
855 (Proposition (\ref{diff0prop})). |
877 (Proposition (\ref{diff0prop})). |
856 For any splitting $X = X_1 \cup X_2$, the following diagram commutes |
878 For any splitting $X = X_1 \cup X_2$, the following diagram commutes |
857 \eq{ \xymatrix{ |
879 \eq{ \xymatrix{ |
858 CD_*(X) \otimes \bc_*(X) \ar[r]^{e_X} & \bc_*(X) \\ |
880 CD_*(X) \otimes \bc_*(X) \ar[r]^{e_X} & \bc_*(X) \\ |
859 CD_*(X_1) \otimes CD_*(X_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
881 CD_*(X_1) \otimes CD_*(X_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
860 \ar@/_4ex/[r]_{e_{X_1} \otimes e_{X_2}} \ar[u]^{\gl \otimes \gl} & |
882 \ar@/_4ex/[r]_{e_{X_1} \otimes e_{X_2}} \ar[u]^{\gl \otimes \gl} & |
861 \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl} |
883 \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl} |
862 } } |
884 } } |
863 Any other map satisfying the above two properties is homotopic to $e_X$. |
885 Any other map satisfying the above two properties is homotopic to $e_X$. |
864 \end{prop} |
886 \end{prop} |
865 |
887 |
866 The proof will occupy the remainder of this section. |
888 The proof will occupy the remainder of this section. |
874 |
896 |
875 Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
897 Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
876 A $k$-parameter family of diffeomorphisms $f: P \times X \to X$ is |
898 A $k$-parameter family of diffeomorphisms $f: P \times X \to X$ is |
877 {\it adapted to $\cU$} if there is a factorization |
899 {\it adapted to $\cU$} if there is a factorization |
878 \eq{ |
900 \eq{ |
879 P = P_1 \times \cdots \times P_m |
901 P = P_1 \times \cdots \times P_m |
880 } |
902 } |
881 (for some $m \le k$) |
903 (for some $m \le k$) |
882 and families of diffeomorphisms |
904 and families of diffeomorphisms |
883 \eq{ |
905 \eq{ |
884 f_i : P_i \times X \to X |
906 f_i : P_i \times X \to X |
885 } |
907 } |
886 such that |
908 such that |
887 \begin{itemize} |
909 \begin{itemize} |
888 \item each $f_i(p, \cdot): X \to X$ is supported on some connected $V_i \sub X$; |
910 \item each $f_i(p, \cdot): X \to X$ is supported on some connected $V_i \sub X$; |
889 \item the $V_i$'s are mutually disjoint; |
911 \item the $V_i$'s are mutually disjoint; |
890 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
912 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
891 where $k_i = \dim(P_i)$; and |
913 where $k_i = \dim(P_i)$; and |
892 \item $f(p, \cdot) = f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot) \circ g$ |
914 \item $f(p, \cdot) = f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot) \circ g$ |
893 for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$. |
915 for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$. |
894 \end{itemize} |
916 \end{itemize} |
895 A chain $x \in C_k(\Diff(X))$ is (by definition) adapted to $\cU$ if it is the sum |
917 A chain $x \in C_k(\Diff(X))$ is (by definition) adapted to $\cU$ if it is the sum |
902 |
924 |
903 The proof will be given in Section \ref{fam_diff_sect}. |
925 The proof will be given in Section \ref{fam_diff_sect}. |
904 |
926 |
905 \medskip |
927 \medskip |
906 |
928 |
907 Let $B_1, \ldots, B_m$ be a collection of disjoint balls in $X$ |
929 Let $B_1, \ldots, B_m$ be a collection of disjoint balls in $X$ |
908 (e.g.~the support of a blob diagram). |
930 (e.g.~the support of a blob diagram). |
909 We say that $f:P\times X\to X$ is {\it compatible} with $\{B_j\}$ if |
931 We say that $f:P\times X\to X$ is {\it compatible} with $\{B_j\}$ if |
910 $f$ has support a disjoint collection of balls $D_i \sub X$ and for all $i$ and $j$ |
932 $f$ has support a disjoint collection of balls $D_i \sub X$ and for all $i$ and $j$ |
911 either $B_j \sub D_i$ or $B_j \cap D_i = \emptyset$. |
933 either $B_j \sub D_i$ or $B_j \cap D_i = \emptyset$. |
912 A chain $x \in CD_k(X)$ is compatible with $\{B_j\}$ if it is a sum of singular cells, |
934 A chain $x \in CD_k(X)$ is compatible with $\{B_j\}$ if it is a sum of singular cells, |
913 each of which is compatible. |
935 each of which is compatible. |
914 (Note that we could strengthen the definition of compatibility to incorporate |
936 (Note that we could strengthen the definition of compatibility to incorporate |
915 a factorization condition, similar to the definition of ``adapted to" above. |
937 a factorization condition, similar to the definition of ``adapted to" above. |
916 The weaker definition given here will suffice for our needs below.) |
938 The weaker definition given here will suffice for our needs below.) |
917 |
939 |
918 \begin{cor} \label{extension_lemma_2} |
940 \begin{cor} \label{extension_lemma_2} |
919 Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is compatible with $\{B_j\}$. |
941 Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is compatible with $\{B_j\}$. |
920 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is compatible with $\{B_j\}$. |
942 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is compatible with $\{B_j\}$. |
921 \end{cor} |
943 \end{cor} |
922 \begin{proof} |
944 \begin{proof} |
923 This will follow from Lemma \ref{extension_lemma} for |
945 This will follow from Lemma \ref{extension_lemma} for |
924 appropriate choice of cover $\cU = \{U_\alpha\}$. |
946 appropriate choice of cover $\cU = \{U_\alpha\}$. |
925 Let $U_{\alpha_1}, \ldots, U_{\alpha_k}$ be any $k$ open sets of $\cU$, and let |
947 Let $U_{\alpha_1}, \ldots, U_{\alpha_k}$ be any $k$ open sets of $\cU$, and let |
926 $V_1, \ldots, V_m$ be the connected components of $U_{\alpha_1}\cup\cdots\cup U_{\alpha_k}$. |
948 $V_1, \ldots, V_m$ be the connected components of $U_{\alpha_1}\cup\cdots\cup U_{\alpha_k}$. |
927 Choose $\cU$ fine enough so that there exist disjoint balls $B'_j \sup B_j$ such that for all $i$ and $j$ |
949 Choose $\cU$ fine enough so that there exist disjoint balls $B'_j \sup B_j$ such that for all $i$ and $j$ |
928 either $V_i \sub B'_j$ or $V_i \cap B'_j = \emptyset$. |
950 either $V_i \sub B'_j$ or $V_i \cap B'_j = \emptyset$. |
929 |
951 |
930 Apply Lemma \ref{extension_lemma} first to each singular cell $f_i$ of $\bd x$, |
952 Apply Lemma \ref{extension_lemma} first to each singular cell $f_i$ of $\bd x$, |
931 with the (compatible) support of $f_i$ in place of $X$. |
953 with the (compatible) support of $f_i$ in place of $X$. |
932 This insures that the resulting homotopy $h_i$ is compatible. |
954 This insures that the resulting homotopy $h_i$ is compatible. |
933 Now apply Lemma \ref{extension_lemma} to $x + \sum h_i$. |
955 Now apply Lemma \ref{extension_lemma} to $x + \sum h_i$. |
934 \nn{actually, need to start with the 0-skeleton of $\bd x$, then 1-skeleton, etc.; fix this} |
956 \nn{actually, need to start with the 0-skeleton of $\bd x$, then 1-skeleton, etc.; fix this} |
935 \end{proof} |
957 \end{proof} |
955 later draft} |
977 later draft} |
956 |
978 |
957 \nn{not sure what the best way to deal with boundary is; for now just give main argument, worry |
979 \nn{not sure what the best way to deal with boundary is; for now just give main argument, worry |
958 about boundary later} |
980 about boundary later} |
959 |
981 |
960 Recall that we are given |
982 Recall that we are given |
961 an open cover $\cU = \{U_\alpha\}$ and an |
983 an open cover $\cU = \{U_\alpha\}$ and an |
962 $x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$. |
984 $x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$. |
963 We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
985 We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
964 |
986 |
965 Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$. |
987 Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$. |
966 |
988 |
967 As a first approximation to the argument we will eventually make, let's replace $x$ |
989 As a first approximation to the argument we will eventually make, let's replace $x$ |
968 with a single singular cell |
990 with a single singular cell |
969 \eq{ |
991 \eq{ |
970 f: P \times X \to X . |
992 f: P \times X \to X . |
971 } |
993 } |
972 Also, we'll ignore for now issues around $\bd P$. |
994 Also, we'll ignore for now issues around $\bd P$. |
973 |
995 |
974 Our homotopy will have the form |
996 Our homotopy will have the form |
975 \eqar{ |
997 \eqar{ |
976 F: I \times P \times X &\to& X \\ |
998 F: I \times P \times X &\to& X \\ |
977 (t, p, x) &\mapsto& f(u(t, p, x), x) |
999 (t, p, x) &\mapsto& f(u(t, p, x), x) |
978 } |
1000 } |
979 for some function |
1001 for some function |
980 \eq{ |
1002 \eq{ |
981 u : I \times P \times X \to P . |
1003 u : I \times P \times X \to P . |
982 } |
1004 } |
983 First we describe $u$, then we argue that it does what we want it to do. |
1005 First we describe $u$, then we argue that it does what we want it to do. |
984 |
1006 |
985 For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$. |
1007 For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$. |
986 The various $K_\alpha$ should be in general position with respect to each other. |
1008 The various $K_\alpha$ should be in general position with respect to each other. |
1005 To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s |
1027 To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s |
1006 which contain $d$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$. |
1028 which contain $d$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$. |
1007 |
1029 |
1008 For $p \in D$ we define |
1030 For $p \in D$ we define |
1009 \eq{ |
1031 \eq{ |
1010 u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} . |
1032 u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} . |
1011 } |
1033 } |
1012 (Recall that $P$ is a single linear cell, so the weighted average of points of $P$ |
1034 (Recall that $P$ is a single linear cell, so the weighted average of points of $P$ |
1013 makes sense.) |
1035 makes sense.) |
1014 |
1036 |
1015 So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$. |
1037 So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$. |
1016 For handles of $J$ of index less than $k$, we will define $u$ to |
1038 For handles of $J$ of index less than $k$, we will define $u$ to |
1017 interpolate between the values on $k$-handles defined above. |
1039 interpolate between the values on $k$-handles defined above. |
1018 |
1040 |
1019 If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate |
1041 If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate |
1020 of $E$. |
1042 of $E$. |
1021 In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$ |
1043 In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$ |
1022 with a $k$-handle. |
1044 with a $k$-handle. |
1023 Let $\beta$ be the index of the $K_\beta$ containing the $k{-}1$-cell |
1045 Let $\beta$ be the index of the $K_\beta$ containing the $k{-}1$-cell |
1024 corresponding to $E$. |
1046 corresponding to $E$. |
1025 Let $q_0, q_1 \in P$ be the points associated to the two $k$-cells of $K_\beta$ |
1047 Let $q_0, q_1 \in P$ be the points associated to the two $k$-cells of $K_\beta$ |
1026 adjacent to the $k{-}1$-cell corresponding to $E$. |
1048 adjacent to the $k{-}1$-cell corresponding to $E$. |
1027 For $p \in E$, define |
1049 For $p \in E$, define |
1028 \eq{ |
1050 \eq{ |
1029 u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha} |
1051 u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha} |
1030 + r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) . |
1052 + r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) . |
1031 } |
1053 } |
1032 |
1054 |
1033 In general, for $E$ a $k{-}j$-handle, there is a normal coordinate |
1055 In general, for $E$ a $k{-}j$-handle, there is a normal coordinate |
1034 $\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron. |
1056 $\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron. |
1035 The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$. |
1057 The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$. |
1038 Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets |
1060 Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets |
1039 the $k{-}j$-cell corresponding to $E$. |
1061 the $k{-}j$-cell corresponding to $E$. |
1040 For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells. |
1062 For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells. |
1041 Now define, for $p \in E$, |
1063 Now define, for $p \in E$, |
1042 \eq{ |
1064 \eq{ |
1043 u(t, p, x) = (1-t)p + t \left( |
1065 u(t, p, x) = (1-t)p + t \left( |
1044 \sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha} |
1066 \sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha} |
1045 + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) |
1067 + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) |
1046 \right) . |
1068 \right) . |
1047 } |
1069 } |
1048 Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension |
1070 Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension |
1049 mentioned above. |
1071 mentioned above. |
1050 |
1072 |
1051 This completes the definition of $u: I \times P \times X \to P$. |
1073 This completes the definition of $u: I \times P \times X \to P$. |
1060 Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions, |
1082 Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions, |
1061 then $F$ is a homotopy through diffeomorphisms. |
1083 then $F$ is a homotopy through diffeomorphisms. |
1062 We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. |
1084 We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. |
1063 We have |
1085 We have |
1064 \eq{ |
1086 \eq{ |
1065 % \pd{F}{x}(t, p, x) = \pd{f}{x}(u(t, p, x), x) + \pd{f}{p}(u(t, p, x), x) \pd{u}{x}(t, p, x) . |
1087 % \pd{F}{x}(t, p, x) = \pd{f}{x}(u(t, p, x), x) + \pd{f}{p}(u(t, p, x), x) \pd{u}{x}(t, p, x) . |
1066 \pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} . |
1088 \pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} . |
1067 } |
1089 } |
1068 Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and |
1090 Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and |
1069 \nn{bounded away from zero, or something like that}. |
1091 \nn{bounded away from zero, or something like that}. |
1070 (Recall that $X$ and $P$ are compact.) |
1092 (Recall that $X$ and $P$ are compact.) |
1071 Also, $\pd{f}{p}$ is bounded. |
1093 Also, $\pd{f}{p}$ is bounded. |
1081 Next we show that for each handle $D \sub P$, $F(1, \cdot, \cdot) : D\times X \to X$ |
1103 Next we show that for each handle $D \sub P$, $F(1, \cdot, \cdot) : D\times X \to X$ |
1082 is a singular cell adapted to $\cU$. |
1104 is a singular cell adapted to $\cU$. |
1083 This will complete the proof of the lemma. |
1105 This will complete the proof of the lemma. |
1084 \nn{except for boundary issues and the `$P$ is a cell' assumption} |
1106 \nn{except for boundary issues and the `$P$ is a cell' assumption} |
1085 |
1107 |
1086 Let $j$ be the codimension of $D$. |
1108 Let $j$ be the codimension of $D$. |
1087 (Or rather, the codimension of its corresponding cell. From now on we will not make a distinction |
1109 (Or rather, the codimension of its corresponding cell. From now on we will not make a distinction |
1088 between handle and corresponding cell.) |
1110 between handle and corresponding cell.) |
1089 Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$, |
1111 Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$, |
1090 where the $j_i$'s are the codimensions of the $K_\alpha$ |
1112 where the $j_i$'s are the codimensions of the $K_\alpha$ |
1091 cells of codimension greater than 0 which intersect to form $D$. |
1113 cells of codimension greater than 0 which intersect to form $D$. |
1108 It follows that the singular cell $D \to \Diff(X)$ can be written as a product |
1130 It follows that the singular cell $D \to \Diff(X)$ can be written as a product |
1109 of a constant map $D' \to \Diff(X)$ and a singular 1-cell $I \to \Diff(X)$. |
1131 of a constant map $D' \to \Diff(X)$ and a singular 1-cell $I \to \Diff(X)$. |
1110 The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set. |
1132 The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set. |
1111 |
1133 |
1112 Next case: $j=2$, $m=1$, $j_1 = 2$. |
1134 Next case: $j=2$, $m=1$, $j_1 = 2$. |
1113 This is similar to the previous case, except that the normal bundle is 2-dimensional instead of |
1135 This is similar to the previous case, except that the normal bundle is 2-dimensional instead of |
1114 1-dimensional. |
1136 1-dimensional. |
1115 We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell |
1137 We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell |
1116 and a 2-cell with support $U_\beta$. |
1138 and a 2-cell with support $U_\beta$. |
1117 |
1139 |
1118 Next case: $j=2$, $m=2$, $j_1 = j_2 = 1$. |
1140 Next case: $j=2$, $m=2$, $j_1 = j_2 = 1$. |
1134 |
1156 |
1135 |
1157 |
1136 \section{$A_\infty$ action on the boundary} |
1158 \section{$A_\infty$ action on the boundary} |
1137 |
1159 |
1138 Let $Y$ be an $n{-}1$-manifold. |
1160 Let $Y$ be an $n{-}1$-manifold. |
1139 The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary |
1161 The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary |
1140 conditions on $\bd(Y\times I) = Y\times \{0\} \cup Y\times\{1\}$, has the structure |
1162 conditions on $\bd(Y\times I) = Y\times \{0\} \cup Y\times\{1\}$, has the structure |
1141 of an $A_\infty$ category. |
1163 of an $A_\infty$ category. |
1142 |
1164 |
1143 Composition of morphisms (multiplication) depends of a choice of homeomorphism |
1165 Composition of morphisms (multiplication) depends of a choice of homeomorphism |
1144 $I\cup I \cong I$. Given this choice, gluing gives a map |
1166 $I\cup I \cong I$. Given this choice, gluing gives a map |
1145 \eq{ |
1167 \eq{ |
1146 \bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c) |
1168 \bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c) |
1147 \cong \bc_*(Y\times I; a, c) |
1169 \cong \bc_*(Y\times I; a, c) |
1148 } |
1170 } |
1149 Using (\ref{CDprop}) and the inclusion $\Diff(I) \sub \Diff(Y\times I)$ gives the various |
1171 Using (\ref{CDprop}) and the inclusion $\Diff(I) \sub \Diff(Y\times I)$ gives the various |
1150 higher associators of the $A_\infty$ structure, more or less canonically. |
1172 higher associators of the $A_\infty$ structure, more or less canonically. |
1151 |
1173 |
1152 \nn{is this obvious? does more need to be said?} |
1174 \nn{is this obvious? does more need to be said?} |
1153 |
1175 |
1154 Let $\cA(Y)$ denote the $A_\infty$ category $\bc_*(Y\times I; \cdot, \cdot)$. |
1176 Let $\cA(Y)$ denote the $A_\infty$ category $\bc_*(Y\times I; \cdot, \cdot)$. |
1155 |
1177 |
1156 Similarly, if $Y \sub \bd X$, a choice of collaring homeomorphism |
1178 Similarly, if $Y \sub \bd X$, a choice of collaring homeomorphism |
1157 $(Y\times I) \cup_Y X \cong X$ gives the collection of complexes $\bc_*(X; r, a)$ |
1179 $(Y\times I) \cup_Y X \cong X$ gives the collection of complexes $\bc_*(X; r, a)$ |
1158 (variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the |
1180 (variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the |
1159 $A_\infty$ category $\{\bc_*(Y\times I; \cdot, \cdot)\}$. |
1181 $A_\infty$ category $\{\bc_*(Y\times I; \cdot, \cdot)\}$. |
1160 Again the higher associators come from the action of $\Diff(I)$ on a collar neighborhood |
1182 Again the higher associators come from the action of $\Diff(I)$ on a collar neighborhood |
1161 of $Y$ in $X$. |
1183 of $Y$ in $X$. |
1162 |
1184 |
1163 In the next section we use the above $A_\infty$ actions to state and prove |
1185 In the next section we use the above $A_\infty$ actions to state and prove |
1174 Let $Y$ be an $n{-}1$-manifold and let $X$ be an $n$-manifold with a copy |
1196 Let $Y$ be an $n{-}1$-manifold and let $X$ be an $n$-manifold with a copy |
1175 of $Y \du -Y$ contained in its boundary. |
1197 of $Y \du -Y$ contained in its boundary. |
1176 Gluing the two copies of $Y$ together we obtain a new $n$-manifold $X\sgl$. |
1198 Gluing the two copies of $Y$ together we obtain a new $n$-manifold $X\sgl$. |
1177 We wish to describe the blob complex of $X\sgl$ in terms of the blob complex |
1199 We wish to describe the blob complex of $X\sgl$ in terms of the blob complex |
1178 of $X$. |
1200 of $X$. |
1179 More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$, |
1201 More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$, |
1180 where $c\sgl \in \cC(\bd X\sgl)$, |
1202 where $c\sgl \in \cC(\bd X\sgl)$, |
1181 in terms of the collection $\{\bc_*(X; c, \cdot, \cdot)\}$, thought of as a representation |
1203 in terms of the collection $\{\bc_*(X; c, \cdot, \cdot)\}$, thought of as a representation |
1182 of the $A_\infty$ category $\cA(Y\du-Y) \cong \cA(Y)\times \cA(Y)\op$. |
1204 of the $A_\infty$ category $\cA(Y\du-Y) \cong \cA(Y)\times \cA(Y)\op$. |
1183 |
1205 |
1184 \begin{thm} |
1206 \begin{thm} |
1185 $\bc_*(X\sgl; c\sgl)$ is quasi-isomorphic to the the self tensor product |
1207 $\bc_*(X\sgl; c\sgl)$ is quasi-isomorphic to the the self tensor product |
1186 of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$. |
1208 of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$. |
1187 \end{thm} |
1209 \end{thm} |
1188 |
1210 |
1189 The proof will occupy the remainder of this section. |
1211 The proof will occupy the remainder of this section. |
1190 |
1212 |
1191 \nn{...} |
1213 \nn{...} |
1198 |
1220 |
1199 |
1221 |
1200 |
1222 |
1201 \section{Extension to ...} |
1223 \section{Extension to ...} |
1202 |
1224 |
1203 \nn{Need to let the input $n$-category $C$ be a graded thing |
1225 \nn{Need to let the input $n$-category $C$ be a graded thing |
1204 (e.g.~DGA or $A_\infty$ $n$-category).} |
1226 (e.g.~DGA or $A_\infty$ $n$-category).} |
1205 |
1227 |
1206 \nn{maybe this should be done earlier in the exposition? |
1228 \nn{maybe this should be done earlier in the exposition? |
1207 if we can plausibly claim that the various proofs work almost |
1229 if we can plausibly claim that the various proofs work almost |
1208 the same with the extended def, then maybe it's better to extend late (here)} |
1230 the same with the extended def, then maybe it's better to extend late (here)} |
1231 |
1253 |
1232 %Recall that for $n$-category picture fields there is an evaluation map |
1254 %Recall that for $n$-category picture fields there is an evaluation map |
1233 %$m: \bc_0(B^n; c, c') \to \mor(c, c')$. |
1255 %$m: \bc_0(B^n; c, c') \to \mor(c, c')$. |
1234 %If we regard $\mor(c, c')$ as a complex concentrated in degree 0, then this becomes a chain |
1256 %If we regard $\mor(c, c')$ as a complex concentrated in degree 0, then this becomes a chain |
1235 %map $m: \bc_*(B^n; c, c') \to \mor(c, c')$. |
1257 %map $m: \bc_*(B^n; c, c') \to \mor(c, c')$. |
1236 |
|
1237 |
|
1238 |