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13 When we need to distinguish between the new and old definitions, we will refer to the |
13 When we need to distinguish between the new and old definitions, we will refer to the |
14 new-fangled and old-fashioned blob complex. |
14 new-fangled and old-fashioned blob complex. |
15 |
15 |
16 \medskip |
16 \medskip |
17 |
17 |
18 \subsection{The small blob complex} |
18 An important technical tool in the proofs of this section is provided by the idea of `small blobs'. |
19 |
19 Fix $\cU$, an open cover of $M$. 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 \input{text/smallblobs} |
20 \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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21 If field have potentially large coupons/boxes, then this is a non-trivial constraint. |
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22 On the other hand, we could probably get away with ignoring this point. |
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23 Maybe the exposition will be better if we sweep this technical detail under the rug?} |
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24 |
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25 \begin{thm}[Small blobs] \label{thm:small-blobs} |
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26 The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. |
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27 \end{thm} |
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28 The proof appears in \S \ref{appendix:small-blobs}. |
21 |
29 |
22 \subsection{A product formula} |
30 \subsection{A product formula} |
23 \label{ss:product-formula} |
31 \label{ss:product-formula} |
24 |
32 |
25 \noop{ |
33 \noop{ |