equal
deleted
inserted
replaced
759 \bc_*(X\bigcup_Y \selfarrow) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
759 \bc_*(X\bigcup_Y \selfarrow) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
760 \end{equation*} |
760 \end{equation*} |
761 \end{itemize} |
761 \end{itemize} |
762 \end{thm} |
762 \end{thm} |
763 |
763 |
|
764 \begin{proof} (Sketch.) |
|
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}. |
|
767 |
|
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 $Y$. |
|
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. |
|
772 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. |
|
774 \end{proof} |
764 |
775 |
765 We next describe the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as above. |
776 We next describe the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as above. |
766 |
777 |
767 \begin{thm}[Product formula] |
778 \begin{thm}[Product formula] |
768 \label{thm:product} |
779 \label{thm:product} |
775 \] |
786 \] |
776 \end{thm} |
787 \end{thm} |
777 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
788 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
778 (see \cite[\S7.1]{1009.5025}). |
789 (see \cite[\S7.1]{1009.5025}). |
779 |
790 |
780 \nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.} |
791 \begin{proof} (Sketch.) |
|
792 |
|
793 \end{proof} |
|
794 |
|
795 %\nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.} |
781 |
796 |
782 \section{Higher Deligne conjecture} |
797 \section{Higher Deligne conjecture} |
783 \label{sec:applications} |
798 \label{sec:applications} |
784 |
799 |
785 \begin{thm}[Higher dimensional Deligne conjecture] |
800 \begin{thm}[Higher dimensional Deligne conjecture] |