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47 In this subsection we define a subcomplex (small blobs) and supercomplex (families of blobs) |
47 In this subsection we define a subcomplex (small blobs) and supercomplex (families of blobs) |
48 of the blob complex, and show that they are both homotopy equivalent to $\bc_*(X)$. |
48 of the blob complex, and show that they are both homotopy equivalent to $\bc_*(X)$. |
49 |
49 |
50 \medskip |
50 \medskip |
51 |
51 |
52 If $b$ is a blob diagram in $\bc_*(X)$, define the {\it support} of $b$, denoted |
52 If $b$ is a blob diagram in $\bc_*(X)$, recall from \S \ref{sec:basic-properties} that the {\it support} of $b$, denoted |
53 $\supp(b)$ or $|b|$, to be the union of the blobs of $b$. |
53 $\supp(b)$ or $|b|$, to be the union of the blobs of $b$. |
54 %For a general $k$-chain $a\in \bc_k(X)$, define the support of $a$ to be the union |
54 %For a general $k$-chain $a\in \bc_k(X)$, define the support of $a$ to be the union |
55 %of the supports of the blob diagrams which appear in it. |
55 %of the supports of the blob diagrams which appear in it. |
56 More generally, we say that a chain $a\in \bc_k(X)$ is supported on $S$ if |
56 More generally, we say that a chain $a\in \bc_k(X)$ is supported on $S$ if |
57 $a = a'\bullet r$, where $a'\in \bc_k(S)$ and $r\in \bc_0(X\setmin S)$. |
57 $a = a'\bullet r$, where $a'\in \bc_k(S)$ and $r\in \bc_0(X\setmin S)$. |