# HG changeset patch # User kevin@6e1638ff-ae45-0410-89bd-df963105f760 # Date 1256697869 0 # Node ID e1d24be683bbfc9aeddf783f025a7761b29aec46 # Parent e0b304e6b9753d83ff86544f8f084e4adc895945 ... diff -r e0b304e6b975 -r e1d24be683bb text/a_inf_blob.tex --- a/text/a_inf_blob.tex Wed Oct 28 00:54:35 2009 +0000 +++ b/text/a_inf_blob.tex Wed Oct 28 02:44:29 2009 +0000 @@ -2,6 +2,7 @@ \section{The blob complex for $A_\infty$ $n$-categories} \label{sec:ainfblob} +\label{sec:gluing} Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we define the blob complex $\bc_*(M)$ to the be the colimit $\cC(M)$ of Section \ref{sec:ncats}. diff -r e0b304e6b975 -r e1d24be683bb text/basic_properties.tex --- a/text/basic_properties.tex Wed Oct 28 00:54:35 2009 +0000 +++ b/text/basic_properties.tex Wed Oct 28 02:44:29 2009 +0000 @@ -27,7 +27,7 @@ we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$ of the quotient map $p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$. -For example, this is always the case if you coefficient ring is a field. +For example, this is always the case if the coefficient ring is a field. Then \begin{prop} \label{bcontract} For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$ @@ -66,14 +66,14 @@ \begin{prop} For fixed fields ($n$-cat), $\bc_*$ is a functor from the category -of $n$-manifolds and diffeomorphisms to the category of chain complexes and +of $n$-manifolds and homeomorphisms to the category of chain complexes and (chain map) isomorphisms. \qed \end{prop} In particular, \begin{prop} \label{diff0prop} -There is an action of $\Diff(X)$ on $\bc_*(X)$. +There is an action of $\Homeo(X)$ on $\bc_*(X)$. \qed \end{prop} @@ -106,16 +106,16 @@ The above map is very far from being an isomorphism, even on homology. This will be fixed in Section \ref{sec:gluing} below. -\nn{Next para not need, since we already use bullet = gluing notation above(?)} +%\nn{Next para not needed, since we already use bullet = gluing notation above(?)} -An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$ -and $X\sgl = X_1 \cup_Y X_2$. -(Typically one of $X_1$ or $X_2$ is a disjoint union of balls.) -For $x_i \in \bc_*(X_i)$, we introduce the notation -\eq{ - x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) . -} -Note that we have resumed our habit of omitting boundary labels from the notation. +%An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$ +%and $X\sgl = X_1 \cup_Y X_2$. +%(Typically one of $X_1$ or $X_2$ is a disjoint union of balls.) +%For $x_i \in \bc_*(X_i)$, we introduce the notation +%\eq{ +% x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) . +%} +%Note that we have resumed our habit of omitting boundary labels from the notation. diff -r e0b304e6b975 -r e1d24be683bb text/definitions.tex --- a/text/definitions.tex Wed Oct 28 00:54:35 2009 +0000 +++ b/text/definitions.tex Wed Oct 28 02:44:29 2009 +0000 @@ -105,10 +105,21 @@ covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. \end{enumerate} -\nn{need to introduce two notations for glued fields --- $x\bullet y$ and $x\sgl$} +There are two notations we commonly use for gluing. +One is +\[ + x\sgl \deq \gl(x) \in \cC(X\sgl) , +\] +for $x\in\cC(X)$. +The other is +\[ + x_1\bullet x_2 \deq \gl(x_1\otimes x_2) \in \cC(X\sgl) , +\] +in the case that $X = X_1 \du X_2$, with $x_i \in \cC(X_i)$. -\bigskip -Using the functoriality and $\bullet\times I$ properties above, together +\medskip + +Using the functoriality and $\cdot\times I$ properties above, together with boundary collar homeomorphisms of manifolds, we can define the notion of {\it extended isotopy}. Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold diff -r e0b304e6b975 -r e1d24be683bb text/evmap.tex --- a/text/evmap.tex Wed Oct 28 00:54:35 2009 +0000 +++ b/text/evmap.tex Wed Oct 28 02:44:29 2009 +0000 @@ -5,7 +5,7 @@ Let $CD_*(X, Y)$ denote $C_*(\Diff(X \to Y))$, the singular chain complex of the space of diffeomorphisms -\nn{or homeomorphisms} +\nn{or homeomorphisms; need to fix the diff vs homeo inconsistency} between the $n$-manifolds $X$ and $Y$ (extending a fixed diffeomorphism $\bd X \to \bd Y$). For convenience, we will permit the singular cells generating $CD_*(X, Y)$ to be more general than simplices --- they can be based on any linear polyhedron. @@ -19,14 +19,22 @@ } On $CD_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X, Y)$ on $\bc_*(X)$ (Proposition (\ref{diff0prop})). -For any splittings $X = X_1 \cup X_2$ and $Y = Y_1 \cup Y_2$, +For any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, the following diagram commutes up to homotopy \eq{ \xymatrix{ - CD_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_{XY}} & \bc_*(Y) \\ - CD_*(X_1, Y_1) \otimes CD_*(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} + CD_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]^(.7){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) \\ + CD_*(X, Y) \otimes \bc_*(X) + \ar@/_4ex/[r]_{e_{XY}} \ar[u]^{\gl \otimes \gl} & + \bc_*(Y) \ar[u]_{\gl} } } +%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{ +% CD_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_{XY}} & \bc_*(Y) \\ +% CD_*(X_1, Y_1) \otimes CD_*(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{prop} diff -r e0b304e6b975 -r e1d24be683bb text/hochschild.tex --- a/text/hochschild.tex Wed Oct 28 00:54:35 2009 +0000 +++ b/text/hochschild.tex Wed Oct 28 02:44:29 2009 +0000 @@ -3,11 +3,16 @@ \section{Hochschild homology when $n=1$} \label{sec:hochschild} +So far we have provided no evidence that blob homology is interesting in degrees +greater than zero. In this section we analyze the blob complex in dimension $n=1$ and find that for $S^1$ the blob complex is homotopy equivalent to the Hochschild complex of the category (algebroid) that we started with. +Thus the blob complex is a natural generalization of something already +known to be interesting in higher homological degrees. -\nn{initial idea for blob complex came from thinking about...} +It is also worth noting that the original idea for the blob complex came from trying +to find a more ``local" description of the Hochschild complex. \nn{need to be consistent about quasi-isomorphic versus homotopy equivalent in this section. @@ -38,7 +43,8 @@ Hochschild complex of $C$. Note that both complexes are free (and hence projective), so it suffices to show that they are quasi-isomorphic. -In order to prove this we will need to extend the blob complex to allow points to also +In order to prove this we will need to extend the +definition of the blob complex to allow points to also be labeled by elements of $C$-$C$-bimodules. Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$. @@ -223,6 +229,7 @@ Suppose we have a short exact sequence of $C$-$C$ bimodules $$\xymatrix{0 \ar[r] & K \ar@{^{(}->}[r]^f & E \ar@{->>}[r]^g & Q \ar[r] & 0}.$$ We'll write $\hat{f}$ and $\hat{g}$ for the image of $f$ and $g$ under the functor. Most of what we need to check is easy. +\nn{don't we need to consider sums here, e.g. $\sum_i(a_i\ot k_i\ot b_i)$ ?} If $(a \tensor k \tensor b) \in \ker(C \tensor K \tensor C \to K)$, to have $\hat{f}(a \tensor k \tensor b) = 0 \in \ker(C \tensor E \tensor C \to E)$, we must have $f(k) = 0 \in E$, which implies $k=0$ itself. If $(a \tensor e \tensor b) \in \ker(C \tensor E \tensor C \to E)$ is in the image of $\ker(C \tensor K \tensor C \to K)$ under $\hat{f}$, clearly $e$ is in the image of the original $f$, so is in the kernel of the original $g$, and so $\hat{g}(a \tensor e \tensor b) = 0$. If $\hat{g}(a \tensor e \tensor b) = 0$, then $g(e) = 0$, so $e = f(\widetilde{e})$ for some $\widetilde{e} \in K$, and $a \tensor e \tensor b = \hat{f}(a \tensor \widetilde{e} \tensor b)$. diff -r e0b304e6b975 -r e1d24be683bb text/ncat.tex --- a/text/ncat.tex Wed Oct 28 00:54:35 2009 +0000 +++ b/text/ncat.tex Wed Oct 28 02:44:29 2009 +0000 @@ -2,14 +2,17 @@ \def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip} -\section{$n$-categories (maybe)} +\section{$n$-categories} \label{sec:ncats} -\nn{experimental section. maybe this should be rolled into other sections. -maybe it should be split off into a separate paper.} +%In order to make further progress establishing properties of the blob complex, +%we need a definition of $A_\infty$ $n$-category that is adapted to our needs. +%(Even in the case $n=1$, we need the new definition given below.) +%It turns out that the $A_\infty$ $n$-category definition and the plain $n$-category +%definition are mostly the same, so we give a new definition of plain +%$n$-categories too. +%We also define modules and tensor products for both plain and $A_\infty$ $n$-categories. -\nn{comment somewhere that what we really need is a convenient def of infty case, including tensor products etc. -but while we're at it might as well do plain case too.} \subsection{Definition of $n$-categories} @@ -18,6 +21,16 @@ (As is the case throughout this paper, by ``$n$-category" we mean a weak $n$-category with strong duality.) +The definitions presented below tie the categories more closely to the topology +and avoid combinatorial questions about, for example, the minimal sufficient +collections of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets. +For examples of topological origin, it is typically easy to show that they +satisfy our axioms. +For examples of a more purely algebraic origin, one would typically need the combinatorial +results that we have avoided here. + +\medskip + Consider first ordinary $n$-categories. We need a set (or sets) of $k$-morphisms for each $0\le k \le n$. We must decide on the ``shape" of the $k$-morphisms. @@ -52,6 +65,7 @@ So we replace the above with \xxpar{Morphisms:} +%\xxpar{Axiom 1 -- Morphisms:} {For each $0 \le k \le n$, we have a functor $\cC_k$ from the category of $k$-balls and homeomorphisms to the category of sets and bijections.} @@ -116,6 +130,7 @@ equipped with an orientation of its once-stabilized tangent bundle. Similarly, in dimension $n-k$ we would have manifolds equipped with an orientation of their $k$ times stabilized tangent bundles. +(cf. [Stolz and Teichner].) Probably should also have a framing of the stabilized dimensions in order to indicate which side the bounded manifold is on. For the moment just stick with unoriented manifolds.}