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16 \medskip |
16 \medskip |
17 |
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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 $M$. |
20 Define the `small blob complex' $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ of all blob diagrams in which every blob is contained in some open set of $\cU$. |
20 Define the `small blob complex' $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ of all blob diagrams in which every blob is contained in some open set of $\cU$, and moreover each field labeling a region cut out by the blobs is splittable into fields on smaller regions, each of which is contained in some open set. |
21 \nn{KW: We need something a little stronger: Every blob diagram (even a 0-blob diagram) is splittable into pieces which are small w.r.t.\ $\cU$. |
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22 If field have potentially large coupons/boxes, then this is a non-trivial constraint. |
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23 On the other hand, we could probably get away with ignoring this point. |
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24 Maybe the exposition will be better if we sweep this technical detail under the rug?} |
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25 |
21 |
26 \begin{thm}[Small blobs] \label{thm:small-blobs} |
22 \begin{thm}[Small blobs] \label{thm:small-blobs} |
27 The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. |
23 The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. |
28 \end{thm} |
24 \end{thm} |
29 The proof appears in \S \ref{appendix:small-blobs}. |
25 The proof appears in \S \ref{appendix:small-blobs}. |