19 |
19 |
20 The most convenient way to prove that maps $e_{XY}$ with the desired properties exist is to |
20 The most convenient way to prove that maps $e_{XY}$ with the desired properties exist is to |
21 introduce a homotopy equivalent alternate version of the blob complex, $\btc_*(X)$, |
21 introduce a homotopy equivalent alternate version of the blob complex, $\btc_*(X)$, |
22 which is more amenable to this sort of action. |
22 which is more amenable to this sort of action. |
23 Recall from Remark \ref{blobsset-remark} that blob diagrams |
23 Recall from Remark \ref{blobsset-remark} that blob diagrams |
24 have the structure of a sort-of-simplicial set. \nn{need a more conventional sounding name: `polyhedral set'?} |
24 have the structure of a cone-product set. |
25 Blob diagrams can also be equipped with a natural topology, which converts this |
25 Blob diagrams can also be equipped with a natural topology, which converts this |
26 sort-of-simplicial set into a sort-of-simplicial space. |
26 cone-product set into a cone-product space. |
27 Taking singular chains of this space we get $\btc_*(X)$. |
27 Taking singular chains of this space we get $\btc_*(X)$. |
28 The details are in \S \ref{ss:alt-def}. |
28 The details are in \S \ref{ss:alt-def}. |
29 We also prove a useful result (Lemma \ref{small-blobs-b}) which says that we can assume that |
29 We also prove a useful result (Lemma \ref{small-blobs-b}) which says that we can assume that |
30 blobs are small with respect to any fixed open cover. |
30 blobs are small with respect to any fixed open cover. |
31 |
31 |
214 \end{proof} |
214 \end{proof} |
215 |
215 |
216 |
216 |
217 \medskip |
217 \medskip |
218 |
218 |
219 Next we define the sort-of-simplicial space version of the blob complex, $\btc_*(X)$. |
219 Next we define the cone-product space version of the blob complex, $\btc_*(X)$. |
220 First we must specify a topology on the set of $k$-blob diagrams, $\BD_k$. |
220 First we must specify a topology on the set of $k$-blob diagrams, $\BD_k$. |
221 We give $\BD_k$ the finest topology such that |
221 We give $\BD_k$ the finest topology such that |
222 \begin{itemize} |
222 \begin{itemize} |
223 \item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous. |
223 \item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous. |
224 \item \nn{don't we need something for collaring maps?} |
224 \item \nn{don't we need something for collaring maps?} |
490 \subsection{[older version still hanging around]} |
490 \subsection{[older version still hanging around]} |
491 \label{ss:old-evmap-remnants} |
491 \label{ss:old-evmap-remnants} |
492 |
492 |
493 \nn{should comment at the start about any assumptions about smooth, PL etc.} |
493 \nn{should comment at the start about any assumptions about smooth, PL etc.} |
494 |
494 |
495 \nn{should maybe mention alternate def of blob complex (sort-of-simplicial space instead of |
495 \nn{should maybe mention alternate def of blob complex (cone-product space instead of |
496 sort-of-simplicial set) where this action would be easy} |
496 cone-product set) where this action would be easy} |
497 |
497 |
498 Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of |
498 Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of |
499 the space of homeomorphisms |
499 the space of homeomorphisms |
500 between the $n$-manifolds $X$ and $Y$ (any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). |
500 between the $n$-manifolds $X$ and $Y$ (any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). |
501 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
501 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |