# HG changeset patch # User Kevin Walker # Date 1270568617 25200 # Node ID cf01e213044a47270855bd01901188dc200f95c5 # Parent 32e75ba211cda4f44fe32f8490fbaa4431b262cf start working on "evaluation map" section diff -r 32e75ba211cd -r cf01e213044a text/evmap.tex --- a/text/evmap.tex Thu Apr 01 15:39:33 2010 -0700 +++ b/text/evmap.tex Tue Apr 06 08:43:37 2010 -0700 @@ -5,7 +5,6 @@ Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of the space of homeomorphisms -\nn{need to fix the diff vs homeo inconsistency} between the $n$-manifolds $X$ and $Y$ (extending a fixed homeomorphism $\bd X \to \bd Y$). For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general than simplices --- they can be based on any linear polyhedron. @@ -15,53 +14,38 @@ \begin{prop} \label{CHprop} For $n$-manifolds $X$ and $Y$ there is a chain map \eq{ - e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) . + e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) } -On $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of $\Homeo(X, Y)$ on $\bc_*(X)$ -(Proposition (\ref{diff0prop})). -For any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, +such that +\begin{enumerate} +\item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of +$\Homeo(X, Y)$ on $\bc_*(X)$ (Proposition (\ref{diff0prop})), and +\item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, the following diagram commutes up to homotopy \eq{ \xymatrix{ - CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]^(.7){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) \\ + CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]^(.7){e_{X\sgl Y\sgl}} \ar[d]^{\gl \otimes \gl} & \bc_*(Y\sgl) \ar[d]_{\gl} \\ CH_*(X, Y) \otimes \bc_*(X) - \ar@/_4ex/[r]_{e_{XY}} \ar[u]^{\gl \otimes \gl} & - \bc_*(Y) \ar[u]_{\gl} + \ar@/_4ex/[r]_{e_{XY}} & + \bc_*(Y) } } -%For any splittings $X = X_1 \cup X_2$ and $Y = Y_1 \cup Y_2$, -%the following diagram commutes up to homotopy -%\eq{ \xymatrix{ -% CH_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_{XY}} & \bc_*(Y) \\ -% CH_*(X_1, Y_1) \otimes CH_*(X_2, Y_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2) -% \ar@/_4ex/[r]_{e_{X_1Y_1} \otimes e_{X_2Y_2}} \ar[u]^{\gl \otimes \gl} & -% \bc_*(Y_1) \otimes \bc_*(Y_2) \ar[u]_{\gl} -%} } -Any other map satisfying the above two properties is homotopic to $e_X$. +\end{enumerate} +Up to (iterated) homotopy, there is a unique family $\{e_{XY}\}$ of chain maps +satisfying the above two conditions. \end{prop} -\nn{need to rewrite for self-gluing instead of gluing two pieces together} - -\nn{Should say something stronger about uniqueness. -Something like: there is -a contractible subcomplex of the complex of chain maps -$CH_*(X) \otimes \bc_*(X) \to \bc_*(X)$ (0-cells are the maps, 1-cells are homotopies, etc.), -and all choices in the construction lie in the 0-cells of this -contractible subcomplex. -Or maybe better to say any two choices are homotopic, and -any two homotopies and second order homotopic, and so on.} \nn{Also need to say something about associativity. Put it in the above prop or make it a separate prop? I lean toward the latter.} \medskip -The proof will occupy the remainder of this section. -\nn{unless we put associativity prop at end} - +The proof will occupy the the next several pages. Without loss of generality, we will assume $X = Y$. \medskip -Let $f: P \times X \to X$ be a family of homeomorphisms and $S \sub X$. +Let $f: P \times X \to X$ be a family of homeomorphisms (e.g. a generator of $CH_*(X)$) +and let $S \sub X$. We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all $x \notin S$ and $p, q \in P$. Equivalently, $f$ is supported on $S$ if there is a family of homeomorphisms $f' : P \times S \to S$ and a `background' homeomorphism $f_0 : X \to X$ so that diff -r 32e75ba211cd -r cf01e213044a text/hochschild.tex --- a/text/hochschild.tex Thu Apr 01 15:39:33 2010 -0700 +++ b/text/hochschild.tex Tue Apr 06 08:43:37 2010 -0700 @@ -467,9 +467,6 @@ \label{fig:hochschild-1-chains} \end{figure} -In degree 2, we send $m\ot a \ot b$ to the sum of 24 ($=6\cdot4$) 2-blob diagrams as shown in -Figure \ref{fig:hochschild-2-chains}. In Figure \ref{fig:hochschild-2-chains} the 1- and 2-blob diagrams are indicated only by their support. -We leave it to the reader to determine the labels of the 1-blob diagrams. \begin{figure}[!ht] \begin{equation*} \mathfig{0.6}{hochschild/2-chains-0} @@ -480,15 +477,6 @@ \caption{The 0-, 1- and 2-chains in the image of $m \tensor a \tensor b$. Only the supports of the 1- and 2-blobs are shown.} \label{fig:hochschild-2-chains} \end{figure} -Each 2-cell in the figure is labeled by a ball $V$ in $S^1$ which contains the support of all -1-blob diagrams in its boundary. -Such a 2-cell corresponds to a sum of the 2-blob diagrams obtained by adding $V$ -as an outer (non-twig) blob to each of the 1-blob diagrams in the boundary of the 2-cell. -Figure \ref{fig:hochschild-example-2-cell} shows this explicitly for the 2-cell -labeled $A$ in Figure \ref{fig:hochschild-2-chains}. -Note that the (blob complex) boundary of this sum of 2-blob diagrams is -precisely the sum of the 1-blob diagrams corresponding to the boundary of the 2-cell. -(Compare with the proof of \ref{bcontract}.) \begin{figure}[!ht] \begin{equation*} @@ -501,3 +489,16 @@ \caption{One of the 2-cells from Figure \ref{fig:hochschild-2-chains}.} \label{fig:hochschild-example-2-cell} \end{figure} + +In degree 2, we send $m\ot a \ot b$ to the sum of 24 ($=6\cdot4$) 2-blob diagrams as shown in +Figure \ref{fig:hochschild-2-chains}. In Figure \ref{fig:hochschild-2-chains} the 1- and 2-blob diagrams are indicated only by their support. +We leave it to the reader to determine the labels of the 1-blob diagrams. +Each 2-cell in the figure is labeled by a ball $V$ in $S^1$ which contains the support of all +1-blob diagrams in its boundary. +Such a 2-cell corresponds to a sum of the 2-blob diagrams obtained by adding $V$ +as an outer (non-twig) blob to each of the 1-blob diagrams in the boundary of the 2-cell. +Figure \ref{fig:hochschild-example-2-cell} shows this explicitly for the 2-cell +labeled $A$ in Figure \ref{fig:hochschild-2-chains}. +Note that the (blob complex) boundary of this sum of 2-blob diagrams is +precisely the sum of the 1-blob diagrams corresponding to the boundary of the 2-cell. +(Compare with the proof of \ref{bcontract}.)