equal
deleted
inserted
replaced
8 We will show below |
8 We will show below |
9 in Corollary \ref{cor:new-old} |
9 in Corollary \ref{cor:new-old} |
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 $\bc_*(M;\cD)$. |
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 $M$. |
31 |
31 |
32 |
32 |
33 Given a system of fields $\cE$ and a $n{-}k$-manifold $F$, recall from |
33 Given a system of fields $\cE$ and a $n{-}k$-manifold $F$, recall from |
34 Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $\cC_F$ |
34 Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $\cC_F$ |
35 defined by $\cC_F(X) = \cE(X\times F)$ if $\dim(X) < k$ and |
35 defined by $\cC_F(X) = \cE(X\times F)$ if $\dim(X) < k$ and |
36 $\cC_F(X) = \bc_*^\cE(X\times F)$ if $\dim(X) = k$. |
36 $\cC_F(X) = \bc_*(X\times F;\cE)$ if $\dim(X) = k$. |
37 |
37 |
38 |
38 |
39 \begin{thm} \label{thm:product} |
39 \begin{thm} \label{thm:product} |
40 Let $Y$ be a $k$-manifold. |
40 Let $Y$ be a $k$-manifold. |
41 Then there is a homotopy equivalence between ``old-fashioned" (blob diagrams) |
41 Then there is a homotopy equivalence between ``old-fashioned" (blob diagrams) |