9 first define it, then show it's hty equivalent to the other def, then observe that |
9 first define it, then show it's hty equivalent to the other def, then observe that |
10 $CH*$ acts. |
10 $CH*$ acts. |
11 maybe salvage some of the original version of this section as a subsection outlining |
11 maybe salvage some of the original version of this section as a subsection outlining |
12 how one might proceed directly.} |
12 how one might proceed directly.} |
13 |
13 |
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14 In this section we extend the action of homeomorphisms on $\bc_*(X)$ |
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15 to an action of {\it families} of homeomorphisms. |
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16 That is, for each pair of homeomorphic manifolds $X$ and $Y$ |
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17 we define a chain map |
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18 \[ |
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19 e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) , |
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20 \] |
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21 where $CH_*(X, Y) = C_*(\Homeo(X, Y))$, the singular chains on the space |
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22 of homeomorphisms from $X$ to $Y$. |
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23 (If $X$ and $Y$ have non-empty boundary, these families of homeomorphisms |
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24 are required to be fixed on the boundaries.) |
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25 See \S \ref{ss:emap-def} for a more precise statement. |
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26 |
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27 The most convenient way to prove that maps $e_{XY}$ with the desired properties exist is to |
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28 introduce a homotopy equivalent alternate version of the blob complex $\btc_*(X)$ |
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29 which is more amenable to this sort of action. |
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30 Recall from Remark \ref{blobsset-remark} that blob diagrams |
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31 have the structure of a sort-of-simplicial set. |
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32 Blob diagrams can also be equipped with a natural topology, which converts this |
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33 sort-of-simplicial set into a sort-of-simplicial space. |
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34 Taking singular chains of this space we get $\btc_*(X)$. |
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35 The details are in \S \ref{ss:alt-def}. |
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36 For technical reasons we also show that requiring the blobs to be |
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37 embedded yields a homotopy equivalent complex. |
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38 |
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39 Since $\bc_*(X)$ and $\btc_*(X)$ are homotopy equivalent one could try to construct |
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40 the $CH_*$ actions directly in terms of $\bc_*(X)$. |
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41 This was our original approach, but working out the details created a nearly unreadable mess. |
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42 We have salvaged a sketch of that approach in \S \ref{ss:old-evmap-remnants}. |
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43 |
14 |
44 |
15 \subsection{Alternative definitions of the blob complex} |
45 \subsection{Alternative definitions of the blob complex} |
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46 \label{ss:alt-def} |
16 |
47 |
17 |
48 |
18 \subsection{Action of \texorpdfstring{$\CH{X}$}{C_*(Homeo(M))}} |
49 \subsection{Action of \texorpdfstring{$\CH{X}$}{C_*(Homeo(M))}} |
19 |
50 \label{ss:emap-def} |
20 |
51 |
21 |
52 |
22 |
53 |
23 \subsection{[older version still hanging around]} |
54 \subsection{[older version still hanging around]} |
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55 \label{ss:old-evmap-remnants} |
24 |
56 |
25 \nn{should comment at the start about any assumptions about smooth, PL etc.} |
57 \nn{should comment at the start about any assumptions about smooth, PL etc.} |
26 |
58 |
27 \nn{should maybe mention alternate def of blob complex (sort-of-simplicial space instead of |
59 \nn{should maybe mention alternate def of blob complex (sort-of-simplicial space instead of |
28 sort-of-simplicial set) where this action would be easy} |
60 sort-of-simplicial set) where this action would be easy} |