710 } |
710 } |
711 \end{equation*} |
711 \end{equation*} |
712 \end{thm} |
712 \end{thm} |
713 |
713 |
714 \begin{proof}(Sketch.) |
714 \begin{proof}(Sketch.) |
715 The most convenient way to prove this is to introduce yet another homotopy equivalent version of |
715 We introduce yet another homotopy equivalent version of |
716 the blob complex, $\cB\cT_*(X)$. |
716 the blob complex, $\cB\cT_*(X)$. |
717 Blob diagrams have a natural topology, which is ignored by $\bc_*(X)$. |
717 Blob diagrams have a natural topology, which is ignored by $\bc_*(X)$. |
718 In $\cB\cT_*(X)$ we take this topology into account, treating the blob diagrams as something |
718 In $\cB\cT_*(X)$ we take this topology into account, treating the blob diagrams as something |
719 analogous to a simplicial space (but with cone-product polyhedra replacing simplices). |
719 analogous to a simplicial space (but with cone-product polyhedra replacing simplices). |
720 More specifically, a generator of $\cB\cT_k(X)$ is an $i$-parameter family of $j$-blob diagrams, with $i+j=k$. |
720 More specifically, a generator of $\cB\cT_k(X)$ is an $i$-parameter family of $j$-blob diagrams, with $i+j=k$. An essential step in the proof of this equivalence is a result to the effect that a $k$-parameter family of homeomorphism can be localized to at most $k$ small sets. |
721 |
721 |
722 With this alternate version in hand, it is straightforward to prove the theorem. |
722 With this alternate version in hand, it is straightforward to prove the theorem. |
723 The evaluation map $\Homeo(X)\times BD_j(X)\to BD_j(X)$ |
723 The evaluation map $\Homeo(X)\times BD_j(X)\to BD_j(X)$ |
724 induces a chain map $\CH{X}\tensor C_*(BD_j(X))\to C_*(BD_j(X))$ |
724 induces a chain map $\CH{X}\tensor C_*(BD_j(X))\to C_*(BD_j(X))$ |
725 and hence a map $e_X: \CH{X} \tensor \cB\cT_*(X) \to \cB\cT_*(X)$. |
725 and hence a map $e_X: \CH{X} \tensor \cB\cT_*(X) \to \cB\cT_*(X)$. |
764 \begin{proof} (Sketch.) |
764 \begin{proof} (Sketch.) |
765 The $A_\infty$ action of $\bc_*(Y)$ follows from the naturality of the blob complex with respect to gluing |
765 The $A_\infty$ action of $\bc_*(Y)$ follows from the naturality of the blob complex with respect to gluing |
766 and the $C_*(\Homeo(-))$ action of Theorem \ref{thm:evaluation}. |
766 and the $C_*(\Homeo(-))$ action of Theorem \ref{thm:evaluation}. |
767 |
767 |
768 Let $T_*$ denote the self tensor product of $\bc_*(X)$, which is a homotopy colimit. |
768 Let $T_*$ denote the self tensor product of $\bc_*(X)$, which is a homotopy colimit. |
769 Let $X_{\mathrm gl}$ denote $X$ glued to itself along $Y$. |
769 There is a tautological map from the 0-simplices of $T_*$ to $\bc_*(X\bigcup_Y \selfarrow)$, |
770 There is a tautological map from the 0-simplices of $T_*$ to $\bc_*(X_{\mathrm gl})$, |
|
771 and this map can be extended to a chain map on all of $T_*$ by sending the higher simplices to zero. |
770 and this map can be extended to a chain map on all of $T_*$ by sending the higher simplices to zero. |
772 Constructing a homotopy inverse to this natural map invloves making various choices, but one can show that the |
771 Constructing a homotopy inverse to this natural map invloves making various choices, but one can show that the |
773 choices form contractible subcomplexes and apply the acyclic models theorem. |
772 choices form contractible subcomplexes and apply the acyclic models theorem. |
774 \end{proof} |
773 \end{proof} |
775 |
774 |