6 anticlimactically tautological definition of the blob |
6 anticlimactically tautological definition of the blob |
7 complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of \S\ref{ss:ncat_fields}. |
7 complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of \S\ref{ss:ncat_fields}. |
8 |
8 |
9 We will show below |
9 We will show below |
10 in Corollary \ref{cor:new-old} |
10 in Corollary \ref{cor:new-old} |
11 that when $\cC$ is obtained from a system of fields $\cD$ |
11 that when $\cC$ is obtained from a system of fields $\cE$ |
12 as the blob complex of an $n$-ball (see Example \ref{ex:blob-complexes-of-balls}), |
12 as the blob complex of an $n$-ball (see Example \ref{ex:blob-complexes-of-balls}), |
13 $\cl{\cC}(M)$ is homotopy equivalent to |
13 $\cl{\cC}(M)$ is homotopy equivalent to |
14 our original definition of the blob complex $\bc_*(M;\cD)$. |
14 our original definition of the blob complex $\bc_*(M;\cE)$. |
15 |
15 |
16 %\medskip |
16 %\medskip |
17 |
17 |
18 %An important technical tool in the proofs of this section is provided by the idea of ``small blobs". |
18 %An important technical tool in the proofs of this section is provided by the idea of ``small blobs". |
19 %Fix $\cU$, an open cover of $M$. |
19 %Fix $\cU$, an open cover of $M$. |
49 \begin{proof} |
49 \begin{proof} |
50 We will use the concrete description of the homotopy colimit from \S\ref{ss:ncat_fields}. |
50 We will use the concrete description of the homotopy colimit from \S\ref{ss:ncat_fields}. |
51 |
51 |
52 First we define a map |
52 First we define a map |
53 \[ |
53 \[ |
54 \psi: \cl{\cC_F}(Y) \to \bc_*(Y\times F;C) . |
54 \psi: \cl{\cC_F}(Y) \to \bc_*(Y\times F;\cE) . |
55 \] |
55 \] |
56 On 0-simplices of the hocolimit |
56 On 0-simplices of the hocolimit |
57 we just glue together the various blob diagrams on $X_i\times F$ |
57 we just glue together the various blob diagrams on $X_i\times F$ |
58 (where $X_i$ is a component of a permissible decomposition of $Y$) to get a blob diagram on |
58 (where $X_i$ is a component of a permissible decomposition of $Y$) to get a blob diagram on |
59 $Y\times F$. |
59 $Y\times F$. |
60 For simplices of dimension 1 and higher we define the map to be zero. |
60 For simplices of dimension 1 and higher we define the map to be zero. |
61 It is easy to check that this is a chain map. |
61 It is easy to check that this is a chain map. |
62 |
62 |
63 In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;C)$ |
63 In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;\cE)$ |
64 and a map |
64 and a map |
65 \[ |
65 \[ |
66 \phi: G_* \to \cl{\cC_F}(Y) . |
66 \phi: G_* \to \cl{\cC_F}(Y) . |
67 \] |
67 \] |
68 |
68 |
69 Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding |
69 Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding |
70 decomposition of $Y\times F$ into the pieces $X_i\times F$. |
70 decomposition of $Y\times F$ into the pieces $X_i\times F$. |
71 |
71 |
72 Let $G_*\sub \bc_*(Y\times F;C)$ be the subcomplex generated by blob diagrams $a$ such that there |
72 Let $G_*\sub \bc_*(Y\times F;\cE)$ be the subcomplex generated by blob diagrams $a$ such that there |
73 exists a decomposition $K$ of $Y$ such that $a$ splits along $K\times F$. |
73 exists a decomposition $K$ of $Y$ such that $a$ splits along $K\times F$. |
74 It follows from Lemma \ref{thm:small-blobs} that $\bc_*(Y\times F; C)$ |
74 It follows from Lemma \ref{thm:small-blobs} that $\bc_*(Y\times F; \cE)$ |
75 is homotopic to a subcomplex of $G_*$. |
75 is homotopic to a subcomplex of $G_*$. |
76 (If the blobs of $a$ are small with respect to a sufficiently fine cover then their |
76 (If the blobs of $a$ are small with respect to a sufficiently fine cover then their |
77 projections to $Y$ are contained in some disjoint union of balls.) |
77 projections to $Y$ are contained in some disjoint union of balls.) |
78 Note that the image of $\psi$ is equal to $G_*$. |
78 Note that the image of $\psi$ is equal to $G_*$. |
79 |
79 |