10 that when $\cC$ is obtained from a system of fields $\cD$ |
10 that when $\cC$ is obtained from a system of fields $\cD$ |
11 as the blob complex of an $n$-ball (see Example \ref{ex:blob-complexes-of-balls}), |
11 as the blob complex of an $n$-ball (see Example \ref{ex:blob-complexes-of-balls}), |
12 $\cl{\cC}(M)$ is homotopy equivalent to |
12 $\cl{\cC}(M)$ is homotopy equivalent to |
13 our original definition of the blob complex $\bc_*^\cD(M)$. |
13 our original definition of the blob complex $\bc_*^\cD(M)$. |
14 |
14 |
15 \medskip |
15 %\medskip |
16 |
16 |
17 An important technical tool in the proofs of this section is provided by the idea of ``small blobs". |
17 %An important technical tool in the proofs of this section is provided by the idea of ``small blobs". |
18 Fix $\cU$, an open cover of $M$. |
18 %Fix $\cU$, an open cover of $M$. |
19 Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ |
19 %Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ |
20 of all blob diagrams in which every blob is contained in some open set of $\cU$, |
20 %of all blob diagrams in which every blob is contained in some open set of $\cU$, |
21 and moreover each field labeling a region cut out by the blobs is splittable |
21 %and moreover each field labeling a region cut out by the blobs is splittable |
22 into fields on smaller regions, each of which is contained in some open set of $\cU$. |
22 %into fields on smaller regions, each of which is contained in some open set of $\cU$. |
23 |
23 % |
24 \begin{thm}[Small blobs] \label{thm:small-blobs} |
24 %\begin{thm}[Small blobs] \label{thm:small-blobs} |
25 The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. |
25 %The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. |
26 \end{thm} |
26 %\end{thm} |
27 The proof appears in \S \ref{appendix:small-blobs}. |
27 %The proof appears in \S \ref{appendix:small-blobs}. |
28 |
28 |
29 \subsection{A product formula} |
29 \subsection{A product formula} |
30 \label{ss:product-formula} |
30 \label{ss:product-formula} |
31 |
31 |
32 |
32 |
67 Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding |
67 Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding |
68 decomposition of $Y\times F$ into the pieces $X_i\times F$. |
68 decomposition of $Y\times F$ into the pieces $X_i\times F$. |
69 |
69 |
70 Let $G_*\sub \bc_*(Y\times F;C)$ be the subcomplex generated by blob diagrams $a$ such that there |
70 Let $G_*\sub \bc_*(Y\times F;C)$ be the subcomplex generated by blob diagrams $a$ such that there |
71 exists a decomposition $K$ of $Y$ such that $a$ splits along $K\times F$. |
71 exists a decomposition $K$ of $Y$ such that $a$ splits along $K\times F$. |
72 It follows from Proposition \ref{thm:small-blobs} that $\bc_*(Y\times F; C)$ |
72 It follows from Lemma \ref{thm:small-blobs} that $\bc_*(Y\times F; C)$ |
73 is homotopic to a subcomplex of $G_*$. |
73 is homotopic to a subcomplex of $G_*$. |
74 (If the blobs of $a$ are small with respect to a sufficiently fine cover then their |
74 (If the blobs of $a$ are small with respect to a sufficiently fine cover then their |
75 projections to $Y$ are contained in some disjoint union of balls.) |
75 projections to $Y$ are contained in some disjoint union of balls.) |
76 Note that the image of $\psi$ is equal to $G_*$. |
76 Note that the image of $\psi$ is equal to $G_*$. |
77 |
77 |
307 by gluing the pieces together to get a blob diagram on $X$. |
307 by gluing the pieces together to get a blob diagram on $X$. |
308 On simplices of dimension 1 and greater $\psi$ is zero. |
308 On simplices of dimension 1 and greater $\psi$ is zero. |
309 |
309 |
310 The image of $\psi$ is the subcomplex $G_*\sub \bc(X)$ generated by blob diagrams which split |
310 The image of $\psi$ is the subcomplex $G_*\sub \bc(X)$ generated by blob diagrams which split |
311 over some decomposition of $J$. |
311 over some decomposition of $J$. |
312 It follows from Proposition \ref{thm:small-blobs} that $\bc_*(X)$ is homotopic to |
312 It follows from Lemma \ref{thm:small-blobs} that $\bc_*(X)$ is homotopic to |
313 a subcomplex of $G_*$. |
313 a subcomplex of $G_*$. |
314 |
314 |
315 Next we define a map $\phi:G_*\to \cT$ using the method of acyclic models. |
315 Next we define a map $\phi:G_*\to \cT$ using the method of acyclic models. |
316 As in the proof of Theorem \ref{thm:product}, we assign to a generator $a$ of $G_*$ |
316 As in the proof of Theorem \ref{thm:product}, we assign to a generator $a$ of $G_*$ |
317 an acyclic subcomplex which is (roughly) $\psi\inv(a)$. |
317 an acyclic subcomplex which is (roughly) $\psi\inv(a)$. |