821 \[ |
821 \[ |
822 \hom_A(\bc_*(M), \bc_*(N)) \ot \bc_*(M) \ot_A \bc_*(R) \to \bc_*(N) \ot_A \bc_*(R) . |
822 \hom_A(\bc_*(M), \bc_*(N)) \ot \bc_*(M) \ot_A \bc_*(R) \to \bc_*(N) \ot_A \bc_*(R) . |
823 \] |
823 \] |
824 We think of this map as being associated to a surgery which cuts $M$ out of $M\cup_E R$ and |
824 We think of this map as being associated to a surgery which cuts $M$ out of $M\cup_E R$ and |
825 replaces it with $N$, yielding $N\cup_E R$. |
825 replaces it with $N$, yielding $N\cup_E R$. |
826 |
826 (This is a more general notion of surgery that usual --- $M$ and $N$ can be any manifolds |
827 Recall (Theorem \ref{thm:evaluation}) that chains on the space of mapping cylinders also act on the |
827 which share a common boundary.) |
|
828 |
|
829 Recall (Theorem \ref{thm:evaluation}) that chains on the space of mapping cylinders also act on the |
828 blob complex. |
830 blob complex. |
829 \nn{...} |
831 An $n$-dimensional surgery cylinder is |
830 |
832 defined to be a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}), |
|
833 modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism. |
|
834 One can associated to this data an $(n{+}1)$-manifold with a foliation by intervals, |
|
835 and the relations we impose correspond to homeomorphisms of the $(n{+}1)$-manifolds |
|
836 which preserve the foliation. |
|
837 |
|
838 Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another. |
|
839 \nn{more to do...} |
831 |
840 |
832 \begin{thm}[Higher dimensional Deligne conjecture] |
841 \begin{thm}[Higher dimensional Deligne conjecture] |
833 \label{thm:deligne} |
842 \label{thm:deligne} |
834 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
843 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
835 Since the little $n{+}1$-balls operad is a suboperad of the $n$-SC operad, |
844 Since the little $n{+}1$-balls operad is a suboperad of the $n$-SC operad, |
836 this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball. |
845 this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball. |
837 \end{thm} |
846 \end{thm} |
838 |
|
839 An $n$-dimensional surgery cylinder is a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}), |
|
840 modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism. |
|
841 Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another. |
|
842 |
847 |
843 By the `blob cochains' of a manifold $X$, we mean the $A_\infty$ maps of $\bc_*(X)$ as a $\bc_*(\bdy X)$ $A_\infty$-module. |
848 By the `blob cochains' of a manifold $X$, we mean the $A_\infty$ maps of $\bc_*(X)$ as a $\bc_*(\bdy X)$ $A_\infty$-module. |
844 |
849 |
845 \begin{proof} |
850 \begin{proof} |
846 We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, |
851 We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, |
974 \caption{A small part of $\cell(W)$.} |
979 \caption{A small part of $\cell(W)$.} |
975 \label{partofJfig} |
980 \label{partofJfig} |
976 \end{figure} |
981 \end{figure} |
977 |
982 |
978 \begin{figure} |
983 \begin{figure} |
979 $$\mathfig{.4}{deligne/manifolds}$$ |
984 %$$\mathfig{.4}{deligne/manifolds}$$ |
|
985 $$\mathfig{.4}{deligne/mapping-cylinders}$$ |
980 \caption{An $n$-dimensional surgery cylinder.}\label{delfig2} |
986 \caption{An $n$-dimensional surgery cylinder.}\label{delfig2} |
981 \end{figure} |
987 \end{figure} |
982 |
988 |
983 |
989 |
984 %% For Tables, put caption above table |
990 %% For Tables, put caption above table |