19 |
19 |
20 The most convenient way to prove that maps $e_{XY}$ with the desired properties exist is to |
20 The most convenient way to prove that maps $e_{XY}$ with the desired properties exist is to |
21 introduce a homotopy equivalent alternate version of the blob complex, $\btc_*(X)$, |
21 introduce a homotopy equivalent alternate version of the blob complex, $\btc_*(X)$, |
22 which is more amenable to this sort of action. |
22 which is more amenable to this sort of action. |
23 Recall from Remark \ref{blobsset-remark} that blob diagrams |
23 Recall from Remark \ref{blobsset-remark} that blob diagrams |
24 have the structure of a sort-of-simplicial set. |
24 have the structure of a sort-of-simplicial set. \nn{need a more conventional sounding name: `polyhedral set'?} |
25 Blob diagrams can also be equipped with a natural topology, which converts this |
25 Blob diagrams can also be equipped with a natural topology, which converts this |
26 sort-of-simplicial set into a sort-of-simplicial space. |
26 sort-of-simplicial set into a sort-of-simplicial space. |
27 Taking singular chains of this space we get $\btc_*(X)$. |
27 Taking singular chains of this space we get $\btc_*(X)$. |
28 The details are in \S \ref{ss:alt-def}. |
28 The details are in \S \ref{ss:alt-def}. |
29 We also prove a useful result (Lemma \ref{small-blobs-b}) which says that we can assume that |
29 We also prove a useful result (Lemma \ref{small-blobs-b}) which says that we can assume that |
68 we say that $f$ is supported on $S\sub M$ if $f(x) = x$ for all $x\in M\setmin S$. |
68 we say that $f$ is supported on $S\sub M$ if $f(x) = x$ for all $x\in M\setmin S$. |
69 |
69 |
70 \medskip |
70 \medskip |
71 |
71 |
72 Fix $\cU$, an open cover of $X$. |
72 Fix $\cU$, an open cover of $X$. |
73 Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(X)$ |
73 Define the ``small blob complex" $\bc^{\cU}_*(X)$ to be the subcomplex of $\bc_*(X)$ |
74 of all blob diagrams in which every blob is contained in some open set of $\cU$, |
74 of all blob diagrams in which every blob is contained in some open set of $\cU$, |
75 and moreover each field labeling a region cut out by the blobs is splittable |
75 and moreover each field labeling a region cut out by the blobs is splittable |
76 into fields on smaller regions, each of which is contained in some open set of $\cU$. |
76 into fields on smaller regions, each of which is contained in some open set of $\cU$. |
77 |
77 |
78 \begin{lemma}[Small blobs] \label{small-blobs-b} \label{thm:small-blobs} |
78 \begin{lemma}[Small blobs] \label{small-blobs-b} \label{thm:small-blobs} |
79 The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. |
79 The inclusion $i: \bc^{\cU}_*(X) \into \bc_*(X)$ is a homotopy equivalence. |
80 \end{lemma} |
80 \end{lemma} |
81 |
81 |
82 \begin{proof} |
82 \begin{proof} |
83 It suffices to show that for any finitely generated pair of subcomplexes |
83 It suffices \nn{why? we should spell this out somewhere} to show that for any finitely generated pair $(C_*, D_*)$, with $D_*$ a subcomplex of $C_*$ such that |
84 \[ |
84 \[ |
85 (C_*, D_*) \sub (\bc_*(X), \sbc_*(X)) |
85 (C_*, D_*) \sub (\bc_*(X), \sbc_*(X)) |
86 \] |
86 \] |
87 we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$ |
87 we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$ |
88 and |
88 and |
90 h\bd(x) + \bd h(x) + x \in \sbc_*(X) |
90 h\bd(x) + \bd h(x) + x \in \sbc_*(X) |
91 \] |
91 \] |
92 for all $x\in C_*$. |
92 for all $x\in C_*$. |
93 |
93 |
94 For simplicity we will assume that all fields are splittable into small pieces, so that |
94 For simplicity we will assume that all fields are splittable into small pieces, so that |
95 $\sbc_0(X) = \bc_0$. |
95 $\sbc_0(X) = \bc_0(X)$. |
96 (This is true for all of the examples presented in this paper.) |
96 (This is true for all of the examples presented in this paper.) |
97 Accordingly, we define $h_0 = 0$. |
97 Accordingly, we define $h_0 = 0$. |
98 |
98 |
99 Next we define $h_1$. |
99 Next we define $h_1$. |
100 Let $b\in C_1$ be a 1-blob diagram. |
100 Let $b\in C_1$ be a 1-blob diagram. |
101 Let $B$ be the blob of $b$. |
101 Let $B$ be the blob of $b$. |
102 We will construct a 1-chain $s(b)\in \sbc_1$ such that $\bd(s(b)) = \bd b$ |
102 We will construct a 1-chain $s(b)\in \sbc_1(X)$ such that $\bd(s(b)) = \bd b$ |
103 and the support of $s(b)$ is contained in $B$. |
103 and the support of $s(b)$ is contained in $B$. |
104 (If $B$ is not embedded in $X$, then we implicitly work in some term of a decomposition |
104 (If $B$ is not embedded in $X$, then we implicitly work in some stage of a decomposition |
105 of $X$ where $B$ is embedded. |
105 of $X$ where $B$ is embedded. |
106 See \ref{defn:configuration} and preceding discussion.) |
106 See Definition \ref{defn:configuration} and preceding discussion.) |
107 It then follows from \ref{disj-union-contract} that we can choose |
107 It then follows from Corollary \ref{disj-union-contract} that we can choose |
108 $h_1(b) \in \bc_1(X)$ such that $\bd(h_1(b)) = s(b) - b$. |
108 $h_1(b) \in \bc_1(X)$ such that $\bd(h_1(b)) = s(b) - b$. |
109 |
109 |
110 Roughly speaking, $s(b)$ consists of a series of 1-blob diagrams implementing a series |
110 Roughly speaking, $s(b)$ consists of a series of 1-blob diagrams implementing a series |
111 of small collar maps, plus a shrunken version of $b$. |
111 of small collar maps, plus a shrunken version of $b$. |
112 The composition of all the collar maps shrinks $B$ to a ball which is small with respect to $\cU$. |
112 The composition of all the collar maps shrinks $B$ to a ball which is small with respect to $\cU$. |
113 |
113 |
114 Let $\cV_1$ be an auxiliary open cover of $X$, subordinate to $\cU$ and |
114 Let $\cV_1$ be an auxiliary open cover of $X$, subordinate to $\cU$ and |
115 also satisfying conditions specified below. |
115 also satisfying conditions specified below. |
116 Let $b = (B, u, r)$, $u = \sum a_i$ be the label of $B$, $a_i\in \bc_0(B)$. |
116 Let $b = (B, u, r)$, with $u = \sum a_i$ the label of $B$, and $a_i\in \bc_0(B)$. |
117 Choose a sequence of collar maps $\bar{f}_j:B\cup\text{collar}\to B$ satisfying conditions which we cannot express |
117 Choose a sequence of collar maps $\bar{f}_j:B\cup\text{collar}\to B$ satisfying conditions which we cannot express |
118 until introducing more notation. |
118 until introducing more notation. \nn{needs some rewriting, I guess} |
119 Let $f_j:B\to B$ be the restriction of $\bar{f}_j$ to $B$; $f_j$ maps $B$ homeomorphically to |
119 Let $f_j:B\to B$ be the restriction of $\bar{f}_j$ to $B$; $f_j$ maps $B$ homeomorphically to |
120 a slightly smaller submanifold of $B$. |
120 a slightly smaller submanifold of $B$. |
121 Let $g_j = f_1\circ f_2\circ\cdots\circ f_j$. |
121 Let $g_j = f_1\circ f_2\circ\cdots\circ f_j$. |
122 Let $g$ be the last of the $g_j$'s. |
122 Let $g$ be the last of the $g_j$'s. |
123 Choose the sequence $\bar{f}_j$ so that |
123 Choose the sequence $\bar{f}_j$ so that |
124 $g(B)$ is contained is an open set of $\cV_1$ and |
124 $g(B)$ is contained is an open set of $\cV_1$ and |
125 $g_{j-1}(|f_j|)$ is also contained is an open set of $\cV_1$. |
125 $g_{j-1}(|f_j|)$ is also contained is an open set of $\cV_1$. |
126 |
126 |
127 There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$ |
127 There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$ |
128 (more specifically, $|c_{ij}| = g_{j-1}(|f_j|)$) |
128 (more specifically, $|c_{ij}| = g_{j-1}(|f_j|)$ \nn{doesn't strictly make any sense}) |
129 and $\bd c_{ij} = g_{j-1}(a_i) - g_{j}(a_i)$. |
129 and $\bd c_{ij} = g_{j-1}(a_i) - g_{j}(a_i)$. |
130 Define |
130 Define |
131 \[ |
131 \[ |
132 s(b) = \sum_{i,j} c_{ij} + g(b) |
132 s(b) = \sum_{i,j} c_{ij} + g(b) |
133 \] |
133 \] |
139 Next we define $h_2$. |
139 Next we define $h_2$. |
140 Let $b\in C_2$ be a 2-blob diagram. |
140 Let $b\in C_2$ be a 2-blob diagram. |
141 Let $B = |b|$, either a ball or a union of two balls. |
141 Let $B = |b|$, either a ball or a union of two balls. |
142 By possibly working in a decomposition of $X$, we may assume that the ball(s) |
142 By possibly working in a decomposition of $X$, we may assume that the ball(s) |
143 of $B$ are disjointly embedded. |
143 of $B$ are disjointly embedded. |
144 We will construct a 2-chain $s(b)\in \sbc_2$ such that |
144 We will construct a 2-chain $s(b)\in \sbc_2(X)$ such that |
145 \[ |
145 \[ |
146 \bd(s(b)) = \bd(h_1(\bd b) + b) = s(\bd b) |
146 \bd(s(b)) = \bd(h_1(\bd b) + b) = s(\bd b) |
147 \] |
147 \] |
148 and the support of $s(b)$ is contained in $B$. |
148 and the support of $s(b)$ is contained in $B$. |
149 It then follows from \ref{disj-union-contract} that we can choose |
149 It then follows from Corollary \ref{disj-union-contract} that we can choose |
150 $h_2(b) \in \bc_2(X)$ such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$. |
150 $h_2(b) \in \bc_2(X)$ such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$. |
151 |
151 |
152 Similarly to the construction of $h_1$ above, |
152 Similarly to the construction of $h_1$ above, |
153 $s(b)$ consists of a series of 2-blob diagrams implementing a series |
153 $s(b)$ consists of a series of 2-blob diagrams implementing a series |
154 of small collar maps, plus a shrunken version of $b$. |
154 of small collar maps, plus a shrunken version of $b$. |
155 The composition of all the collar maps shrinks $B$ to a sufficiently small |
155 The composition of all the collar maps shrinks $B$ to a sufficiently small |
156 disjoint union of balls. |
156 disjoint union of balls. |
157 |
157 |
158 Let $\cV_2$ be an auxiliary open cover of $X$, subordinate to $\cU$ and |
158 Let $\cV_2$ be an auxiliary open cover of $X$, subordinate to $\cU$ and |
159 also satisfying conditions specified below. |
159 also satisfying conditions specified below. \nn{This happens sufficiently far below (i.e. not in this paragraph) that we probably should give better warning.} |
160 As before, choose a sequence of collar maps $f_j$ |
160 As before, choose a sequence of collar maps $f_j$ |
161 such that each has support |
161 such that each has support |
162 contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms |
162 contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms |
163 yields an embedding $g:B\to B$ such that $g(B)$ is contained in an open set of $\cV_1$. |
163 yields an embedding $g:B\to B$ such that $g(B)$ is contained in an open set of $\cV_1$. |
164 Let $g_j:B\to B$ be the embedding at the $j$-th stage. |
164 Let $g_j:B\to B$ be the embedding at the $j$-th stage. |
166 Fix $j$. |
166 Fix $j$. |
167 We will construct a 2-chain $d_j$ such that $\bd d_j = g_{j-1}(s(\bd b)) - g_{j}(s(\bd b))$. |
167 We will construct a 2-chain $d_j$ such that $\bd d_j = g_{j-1}(s(\bd b)) - g_{j}(s(\bd b))$. |
168 Let $s(\bd b) = \sum e_k$, and let $\{p_m\}$ be the 0-blob diagrams |
168 Let $s(\bd b) = \sum e_k$, and let $\{p_m\}$ be the 0-blob diagrams |
169 appearing in the boundaries of the $e_k$. |
169 appearing in the boundaries of the $e_k$. |
170 As in the construction of $h_1$, we can choose 1-blob diagrams $q_m$ such that |
170 As in the construction of $h_1$, we can choose 1-blob diagrams $q_m$ such that |
171 $\bd q_m = g_{j-1}(p_m) - g_j(p_m)$ and $\supp(q_m)$ is contained in an open set of $\cV_1$. |
171 $\bd q_m = g_{j-1}(p_m) - g_j(p_m)$ and $|q_m|$ is contained in an open set of $\cV_1$. |
172 If $x$ is a sum of $p_m$'s, we denote the corresponding sum of $q_m$'s by $q(x)$. |
172 If $x$ is a sum of $p_m$'s, we denote the corresponding sum of $q_m$'s by $q(x)$. |
173 |
173 |
174 Now consider, for each $k$, $g_{j-1}(e_k) - q(\bd e_k)$. |
174 Now consider, for each $k$, $g_{j-1}(e_k) - q(\bd e_k)$. |
175 This is a 1-chain whose boundary is $g_j(\bd e_k)$. |
175 This is a 1-chain whose boundary is $g_j(\bd e_k)$. |
176 The support of $e_k$ is $g_{j-1}(V)$ for some $V\in \cV_1$, and |
176 The support of $e_k$ is $g_{j-1}(V)$ for some $V\in \cV_1$, and |
181 arising in the construction of $h_2$, lies inside a disjoint union of balls $U$ |
181 arising in the construction of $h_2$, lies inside a disjoint union of balls $U$ |
182 such that each individual ball lies in an open set of $\cV_2$. |
182 such that each individual ball lies in an open set of $\cV_2$. |
183 (In this case there are either one or two balls in the disjoint union.) |
183 (In this case there are either one or two balls in the disjoint union.) |
184 For any fixed open cover $\cV_2$ this condition can be satisfied by choosing $\cV_1$ |
184 For any fixed open cover $\cV_2$ this condition can be satisfied by choosing $\cV_1$ |
185 to be a sufficiently fine cover. |
185 to be a sufficiently fine cover. |
186 It follows from \ref{disj-union-contract} that we can choose |
186 It follows from Corollary \ref{disj-union-contract} that we can choose |
187 $x_k \in \bc_2(X)$ with $\bd x_k = g_{j-1}(e_k) - g_j(e_k) - q(\bd e_k)$ |
187 $x_k \in \bc_2(X)$ with $\bd x_k = g_{j-1}(e_k) - g_j(e_k) - q(\bd e_k)$ |
188 and with $\supp(x_k) = U$. |
188 and with $\supp(x_k) = U$. |
189 We can now take $d_j \deq \sum x_k$. |
189 We can now take $d_j \deq \sum x_k$. |
190 It is clear that $\bd d_j = \sum (g_{j-1}(e_k) - g_j(e_k)) = g_{j-1}(s(\bd b)) - g_{j}(s(\bd b))$, as desired. |
190 It is clear that $\bd d_j = \sum (g_{j-1}(e_k) - g_j(e_k)) = g_{j-1}(s(\bd b)) - g_{j}(s(\bd b))$, as desired. |
191 \nn{should maybe have figure} |
191 \nn{should maybe have figure} |
217 Next we define the sort-of-simplicial space version of the blob complex, $\btc_*(X)$. |
217 Next we define the sort-of-simplicial space version of the blob complex, $\btc_*(X)$. |
218 First we must specify a topology on the set of $k$-blob diagrams, $\BD_k$. |
218 First we must specify a topology on the set of $k$-blob diagrams, $\BD_k$. |
219 We give $\BD_k$ the finest topology such that |
219 We give $\BD_k$ the finest topology such that |
220 \begin{itemize} |
220 \begin{itemize} |
221 \item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous. |
221 \item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous. |
|
222 \item \nn{don't we need something for collaring maps?} |
222 \item The gluing maps $\BD_k(M)\to \BD_k(M\sgl)$ are continuous. |
223 \item The gluing maps $\BD_k(M)\to \BD_k(M\sgl)$ are continuous. |
223 \item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous, |
224 \item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous, |
224 where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on |
225 where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on |
225 $\bc_0(B)$ comes from the generating set $\BD_0(B)$. |
226 $\bc_0(B)$ comes from the generating set $\BD_0(B)$. \nn{don't we need to say more to specify a topology on an $\infty$-dimensional vector space} |
226 \end{itemize} |
227 \end{itemize} |
227 |
228 |
228 We can summarize the above by saying that in the typical continuous family |
229 We can summarize the above by saying that in the typical continuous family |
229 $P\to \BD_k(M)$, $p\mapsto \left(B_i(p), u_i(p), r(p)\right)$, $B_i(p)$ and $r(p)$ are induced by a map |
230 $P\to \BD_k(X)$, $p\mapsto \left(B_i(p), u_i(p), r(p)\right)$, $B_i(p)$ and $r(p)$ are induced by a map |
230 $P\to \Homeo(M)$, with the twig blob labels $u_i(p)$ varying independently. |
231 $P\to \Homeo(X)$, with the twig blob labels $u_i(p)$ varying independently. |
231 We note that while we've decided not to allow the blobs $B_i(p)$ to vary independently of the field $r(p)$, |
232 We note that while we've decided not to allow the blobs $B_i(p)$ to vary independently of the field $r(p)$, |
232 if we did allow this it would not affect the truth of the claims we make below. |
233 if we did allow this it would not affect the truth of the claims we make below. |
233 In particular, we would get a homotopy equivalent complex $\btc_*(M)$. |
234 In particular, such a definition of $\btc_*(X)$ would result in a homotopy equivalent complex. |
234 |
235 |
235 Next we define $\btc_*(X)$ to be the total complex of the double complex (denoted $\btc_{**}$) |
236 Next we define $\btc_*(X)$ to be the total complex of the double complex (denoted $\btc_{**}$) |
236 whose $(i,j)$ entry is $C_j(\BD_i)$, the singular $j$-chains on the space of $i$-blob diagrams. |
237 whose $(i,j)$ entry is $C_j(\BD_i)$, the singular $j$-chains on the space of $i$-blob diagrams. |
237 The vertical boundary of the double complex, |
238 The vertical boundary of the double complex, |
238 denoted $\bd_t$, is the singular boundary, and the horizontal boundary, denoted $\bd_b$, is |
239 denoted $\bd_t$, is the singular boundary, and the horizontal boundary, denoted $\bd_b$, is |
239 the blob boundary. |
240 the blob boundary. Following the usual sign convention, we have $\bd = \bd_b + (-1)^i \bd_t$. |
240 |
241 |
241 We will regard $\bc_*(X)$ as the subcomplex $\btc_{*0}(X) \sub \btc_{**}(X)$. |
242 We will regard $\bc_*(X)$ as the subcomplex $\btc_{*0}(X) \sub \btc_{**}(X)$. |
242 The main result of this subsection is |
243 The main result of this subsection is |
243 |
244 |
244 \begin{lemma} \label{lem:bc-btc} |
245 \begin{lemma} \label{lem:bc-btc} |
264 where |
265 where |
265 \[ |
266 \[ |
266 e: \btc_{ij}\to\btc_{i+1,j} |
267 e: \btc_{ij}\to\btc_{i+1,j} |
267 \] |
268 \] |
268 adds an outermost blob, equal to all of $B^n$, to the $j$-parameter family of blob diagrams. |
269 adds an outermost blob, equal to all of $B^n$, to the $j$-parameter family of blob diagrams. |
|
270 Note that for fixed $i$, $e$ is a chain map, i.e. $\bd_t e = e \bd_t$. |
269 |
271 |
270 A generator $y\in \btc_{0j}$ is a map $y:P\to \BD_0$, where $P$ is some $j$-dimensional polyhedron. |
272 A generator $y\in \btc_{0j}$ is a map $y:P\to \BD_0$, where $P$ is some $j$-dimensional polyhedron. |
271 We define $r(y)\in \btc_{0j}$ to be the constant function $r\circ y : P\to \BD_0$. |
273 We define $r(y)\in \btc_{0j}$ to be the constant function $r\circ y : P\to \BD_0$. \nn{I found it pretty confusing to reuse the letter $r$ here.} |
272 Let $c(r(y))\in \btc_{0,j+1}$ be the constant map from the cone of $P$ to $\BD_0$ taking |
274 Let $c(r(y))\in \btc_{0,j+1}$ be the constant map from the cone of $P$ to $\BD_0$ taking |
273 the same value (namely $r(y(p))$, for any $p\in P$). |
275 the same value (namely $r(y(p))$, for any $p\in P$). |
274 Let $e(y - r(y)) \in \btc_{1j}$ denote the $j$-parameter family of 1-blob diagrams |
276 Let $e(y - r(y)) \in \btc_{1j}$ denote the $j$-parameter family of 1-blob diagrams |
275 whose value at $p\in P$ is the blob $B^n$ with label $y(p) - r(y(p))$. |
277 whose value at $p\in P$ is the blob $B^n$ with label $y(p) - r(y(p))$. |
276 Now define, for $y\in \btc_{0j}$, |
278 Now define, for $y\in \btc_{0j}$, |
302 e(\bd_t x - r(\bd_t x)) + c(r(\bd_t x)) \\ |
304 e(\bd_t x - r(\bd_t x)) + c(r(\bd_t x)) \\ |
303 &= x - r(x) + \bd_t(c(r(x))) + c(r(\bd_t x)) \\ |
305 &= x - r(x) + \bd_t(c(r(x))) + c(r(\bd_t x)) \\ |
304 &= x - r(x) + r(x) \\ |
306 &= x - r(x) + r(x) \\ |
305 &= x. |
307 &= x. |
306 \end{align*} |
308 \end{align*} |
|
309 Here we have used the fact that $\bd_b(c(r(x))) = 0$ since $c(r(x))$ is a $0$-blob diagram, as well as that $\bd_t(e(r(x))) = e(r(\bd_t x))$ \nn{explain why this is true?} and $c(r(\bd_t x)) - \bd_t(c(r(x))) = r(x)$ \nn{explain?}. |
|
310 |
307 For $x\in \btc_{00}$ we have |
311 For $x\in \btc_{00}$ we have |
308 \nn{ignoring signs} |
312 \nn{ignoring signs} |
309 \begin{align*} |
313 \begin{align*} |
310 \bd h(x) + h(\bd x) &= \bd_b(e(x - r(x))) + \bd_t(c(r(x))) \\ |
314 \bd h(x) + h(\bd x) &= \bd_b(e(x - r(x))) + \bd_t(c(r(x))) \\ |
311 &= x - r(x) + r(x) - r(x)\\ |
315 &= x - r(x) + r(x) - r(x)\\ |
312 &= x - r(x). |
316 &= x - r(x). \qedhere |
313 \end{align*} |
317 \end{align*} |
314 \end{proof} |
318 \end{proof} |
315 |
319 |
316 \begin{lemma} \label{btc-prod} |
320 \begin{lemma} \label{btc-prod} |
317 For manifolds $X$ and $Y$, we have $\btc_*(X\du Y) \simeq \btc_*(X)\otimes\btc_*(Y)$. |
321 For manifolds $X$ and $Y$, we have $\btc_*(X\du Y) \simeq \btc_*(X)\otimes\btc_*(Y)$. |
318 \end{lemma} |
322 \end{lemma} |
319 \begin{proof} |
323 \begin{proof} |
320 This follows from the Eilenber-Zilber theorem and the fact that |
324 This follows from the Eilenberg-Zilber theorem and the fact that |
321 \[ |
325 \begin{align*} |
322 \BD_k(X\du Y) \cong \coprod_{i+j=k} \BD_i(X)\times\BD_j(Y) . |
326 \BD_k(X\du Y) & \cong \coprod_{i+j=k} \BD_i(X)\times\BD_j(Y) . \qedhere |
323 \] |
327 \end{align*} |
324 \end{proof} |
328 \end{proof} |
325 |
329 |
326 For $S\sub X$, we say that $a\in \btc_k(X)$ is {\it supported on $S$} |
330 For $S\sub X$, we say that $a\in \btc_k(X)$ is {\it supported on $S$} |
327 if there exists $a'\in \btc_k(S)$ |
331 if there exists $a'\in \btc_k(S)$ |
328 and $r\in \btc_0(X\setmin S)$ such that $a = a'\bullet r$. |
332 and $r\in \btc_0(X\setmin S)$ such that $a = a'\bullet r$. |
356 We now apply Lemma \ref{extension_lemma_c} to get families which are supported |
360 We now apply Lemma \ref{extension_lemma_c} to get families which are supported |
357 on balls $D_i$ contained in open sets of $\cU$. |
361 on balls $D_i$ contained in open sets of $\cU$. |
358 \end{proof} |
362 \end{proof} |
359 |
363 |
360 |
364 |
361 \begin{proof}[Proof of \ref{lem:bc-btc}] |
365 \begin{proof}[Proof of Lemma \ref{lem:bc-btc}] |
362 Armed with the above lemmas, we can now proceed similarly to the proof of \ref{small-blobs-b}. |
366 Armed with the above lemmas, we can now proceed similarly to the proof of Lemma \ref{small-blobs-b}. |
363 |
367 |
364 It suffices to show that for any finitely generated pair of subcomplexes |
368 It suffices to show that for any finitely generated pair of subcomplexes |
365 $(C_*, D_*) \sub (\btc_*(X), \bc_*(X))$ |
369 $(C_*, D_*) \sub (\btc_*(X), \bc_*(X))$ |
366 we can find a homotopy $h:C_*\to \btc_*(X)$ such that $h(D_*) \sub \bc_*(X)$ |
370 we can find a homotopy $h:C_*\to \btc_*(X)$ such that $h(D_*) \sub \bc_*(X)$ |
367 and $x + h\bd(x) + \bd h(X) \in \bc_*(X)$ for all $x\in C_*$. |
371 and $x + h\bd(x) + \bd h(x) \in \bc_*(X)$ for all $x\in C_*$. |
368 |
372 |
369 By Lemma \ref{small-top-blobs}, we may assume that $C_* \sub \btc_*^\cU(X)$ for some |
373 By Lemma \ref{small-top-blobs}, we may assume that $C_* \sub \btc_*^\cU(X)$ for some |
370 cover $\cU$ of our choosing. |
374 cover $\cU$ of our choosing. |
371 We choose $\cU$ fine enough so that each generator of $C_*$ is supported on a disjoint union of balls. |
375 We choose $\cU$ fine enough so that each generator of $C_*$ is supported on a disjoint union of balls. |
372 (This is possible since the original $C_*$ was finite and therefore had bounded dimension.) |
376 (This is possible since the original $C_*$ was finite and therefore had bounded dimension.) |
374 Since $\bc_0(X) = \btc_0(X)$, we can take $h_0 = 0$. |
378 Since $\bc_0(X) = \btc_0(X)$, we can take $h_0 = 0$. |
375 |
379 |
376 Let $b \in C_1$ be a generator. |
380 Let $b \in C_1$ be a generator. |
377 Since $b$ is supported in a disjoint union of balls, |
381 Since $b$ is supported in a disjoint union of balls, |
378 we can find $s(b)\in \bc_1$ with $\bd (s(b)) = \bd b$ |
382 we can find $s(b)\in \bc_1$ with $\bd (s(b)) = \bd b$ |
379 (by \ref{disj-union-contract}), and also $h_1(b) \in \btc_2$ |
383 (by Corollary \ref{disj-union-contract}), and also $h_1(b) \in \btc_2(X)$ |
380 such that $\bd (h_1(b)) = s(b) - b$ |
384 such that $\bd (h_1(b)) = s(b) - b$ |
381 (by \ref{bt-contract} and \ref{btc-prod}). |
385 (by Lemmas \ref{bt-contract} and \ref{btc-prod}). |
382 |
386 |
383 Now let $b$ be a generator of $C_2$. |
387 Now let $b$ be a generator of $C_2$. |
384 If $\cU$ is fine enough, there is a disjoint union of balls $V$ |
388 If $\cU$ is fine enough, there is a disjoint union of balls $V$ |
385 on which $b + h_1(\bd b)$ is supported. |
389 on which $b + h_1(\bd b)$ is supported. |
386 Since $\bd(b + h_1(\bd b)) = s(\bd b) \in \bc_2$, we can find |
390 Since $\bd(b + h_1(\bd b)) = s(\bd b) \in \bc_2(X)$, we can find |
387 $s(b)\in \bc_2$ with $\bd(s(b)) = \bd(b + h_1(\bd b))$ (by \ref{disj-union-contract}). |
391 $s(b)\in \bc_2(X)$ with $\bd(s(b)) = \bd(b + h_1(\bd b))$ (by Corollary \ref{disj-union-contract}). |
388 By \ref{bt-contract} and \ref{btc-prod}, we can now find |
392 By Lemmas \ref{bt-contract} and \ref{btc-prod}, we can now find |
389 $h_2(b) \in \btc_3$, also supported on $V$, such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$ |
393 $h_2(b) \in \btc_3(X)$, also supported on $V$, such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$ |
390 |
394 |
391 The general case, $h_k$, is similar. |
395 The general case, $h_k$, is similar. |
392 \end{proof} |
396 \end{proof} |
393 |
397 |
394 The proof of \ref{lem:bc-btc} constructs a homotopy inverse to the inclusion |
398 The proof of Lemma \ref{lem:bc-btc} constructs a homotopy inverse to the inclusion |
395 $\bc_*(X)\sub \btc_*(X)$. |
399 $\bc_*(X)\sub \btc_*(X)$. |
396 One might ask for more: a contractible set of possible homotopy inverses, or at least an |
400 One might ask for more: a contractible set of possible homotopy inverses, or at least an |
397 $m$-connected set for arbitrarily large $m$. |
401 $m$-connected set for arbitrarily large $m$. |
398 The latter can be achieved with finer control over the various |
402 The latter can be achieved with finer control over the various |
399 choices of disjoint unions of balls in the above proofs, but we will not pursue this here. |
403 choices of disjoint unions of balls in the above proofs, but we will not pursue this here. |
438 \end{thm} |
442 \end{thm} |
439 |
443 |
440 \begin{proof} |
444 \begin{proof} |
441 In light of Lemma \ref{lem:bc-btc}, it suffices to prove the theorem with |
445 In light of Lemma \ref{lem:bc-btc}, it suffices to prove the theorem with |
442 $\bc_*$ replaced by $\btc_*$. |
446 $\bc_*$ replaced by $\btc_*$. |
443 And in fact for $\btc_*$ we get a sharper result: we can omit |
447 In fact, for $\btc_*$ we get a sharper result: we can omit |
444 the ``up to homotopy" qualifiers. |
448 the ``up to homotopy" qualifiers. |
445 |
449 |
446 Let $f\in CH_k(X, Y)$, $f:P^k\to \Homeo(X \to Y)$ and $a\in \btc_{ij}(X)$, |
450 Let $f\in CH_k(X, Y)$, $f:P^k\to \Homeo(X \to Y)$ and $a\in \btc_{ij}(X)$, |
447 $a:Q^j \to \BD_i(X)$. |
451 $a:Q^j \to \BD_i(X)$. |
448 Define $e_{XY}(f\ot a)\in \btc_{i,j+k}(Y)$ by |
452 Define $e_{XY}(f\ot a)\in \btc_{i,j+k}(Y)$ by |