# HG changeset patch # User Kevin Walker # Date 1280237138 14400 # Node ID 6ba3a46a0b508ad3d619897fb5ca4c2f520df53f # Parent 7caafccef7e8d19d27865917a89f14a889403020 more revisions of intro diff -r 7caafccef7e8 -r 6ba3a46a0b50 text/intro.tex --- a/text/intro.tex Mon Jul 26 22:57:43 2010 -0400 +++ b/text/intro.tex Tue Jul 27 09:25:38 2010 -0400 @@ -111,7 +111,7 @@ \draw[->] (C) -- node[left=10pt] { Example \ref{ex:traditional-n-categories(fields)} \\ and \S \ref{ss:ncat_fields} - %$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle \cU(B) = \DirectSum_{c \in \cell(B)} \ker \ev: \cC(c) \to \cC(B)$ + %$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle \cU(B) = \DirectSum_{c \in \cell(B)} \ker e: \cC(c) \to \cC(B)$ } (FU); \draw[->] (BC) -- node[left] {$H_0$} node[right] {c.f. Theorem \ref{thm:skein-modules}} (A); @@ -134,12 +134,9 @@ Appendix \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, thought of as a topological $n$-category, in terms of the topology of $M$. -Throughout the paper we typically prefer concrete categories (vector spaces, chain complexes) -\nn{...} - - -%\item related: we are being unsophisticated from a homotopy theory point of -%view and using chain complexes in many places where we could get by with spaces +%%%% this is said later in the intro +%Throughout the paper we typically prefer concrete categories (vector spaces, chain complexes) +%even when we could work in greater generality (symmetric monoidal categories, model categories, etc.). %\item ? one of the points we make (far) below is that there is not really much %difference between (a) systems of fields and local relations and (b) $n$-cats; @@ -151,8 +148,6 @@ \label{sec:motivations} We will briefly sketch our original motivation for defining the blob complex. -\nn{this is adapted from an old draft of the intro; it needs further modification -in order to better integrate it into the current intro.} As a starting point, consider TQFTs constructed via fields and local relations. (See \S\ref{sec:tqftsviafields} or \cite{kw:tqft}.) @@ -166,7 +161,7 @@ It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together with a link $L \subset \bd W$. The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$. -\todo{I'm tempted to replace $A_{Kh}$ with $\cl{Kh}$ throughout this page -S} +%\todo{I'm tempted to replace $A_{Kh}$ with $\cl{Kh}$ throughout this page -S} How would we go about computing $A_{Kh}(W^4, L)$? For the Khovanov homology of a link in $S^3$ the main tool is the exact triangle (long exact sequence) @@ -178,8 +173,8 @@ associated to $B^3$ (with appropriate boundary conditions). The coend is not an exact functor, so the exactness of the triangle breaks. - -The obvious solution to this problem is to replace the coend with its derived counterpart. +The obvious solution to this problem is to replace the coend with its derived counterpart, +Hochschild homology. This presumably works fine for $S^1\times B^3$ (the answer being the Hochschild homology of an appropriate bimodule), but for more complicated 4-manifolds this leaves much to be desired. If we build our manifold up via a handle decomposition, the computation @@ -187,7 +182,9 @@ A different handle decomposition of the same manifold would yield a different sequence of derived coends. To show that our definition in terms of derived coends is well-defined, we -would need to show that the above two sequences of derived coends yield the same answer. +would need to show that the above two sequences of derived coends yield +isomorphic answers, and that the isomorphism does not depend on any +choices we made along the way. This is probably not easy to do. Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$ @@ -201,7 +198,7 @@ \[ \text{linear combinations of fields} \;\big/\; \text{local relations} , \] -with an appropriately free resolution (the ``blob complex") +with an appropriately free resolution (the blob complex) \[ \cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) . \] @@ -210,11 +207,6 @@ $\bc_2$ is linear combinations of relations amongst relations on $W$, and so on. -None of the above ideas depend on the details of the Khovanov homology example, -so we develop the general theory in this paper and postpone specific applications -to later papers. - - \subsection{Formal properties} \label{sec:properties} @@ -236,10 +228,12 @@ The blob complex is also functorial (indeed, exact) with respect to $\cF$, although we will not address this in detail here. +\nn{KW: what exactly does ``exact in $\cF$" mean? +Do we mean a similar statement for module labels?} \begin{property}[Disjoint union] \label{property:disjoint-union} -The blob complex of a disjoint union is naturally the tensor product of the blob complexes. +The blob complex of a disjoint union is naturally isomorphic to the tensor product of the blob complexes. \begin{equation*} \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) \end{equation*} @@ -264,16 +258,19 @@ \begin{property}[Contractibility] \label{property:contractibility}% -With field coefficients, the blob complex on an $n$-ball is contractible in the sense that it is homotopic to its $0$-th homology. -Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces associated by the system of fields $\cF$ to balls. +With field coefficients, the blob complex on an $n$-ball is contractible in the sense +that it is homotopic to its $0$-th homology. +Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces +associated by the system of fields $\cF$ to balls. \begin{equation*} -\xymatrix{\bc_*(B^n;\cF) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cF)) \ar[r]^(0.6)\iso & \cF(B^n)} +\xymatrix{\bc_*(B^n;\cF) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cF)) \ar[r]^(0.6)\iso & A_\cF(B^n)} \end{equation*} \end{property} Properties \ref{property:functoriality} will be immediate from the definition given in \S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there. -Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. +Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and +\ref{property:contractibility} are established in \S \ref{sec:basic-properties}. \subsection{Specializations} \label{sec:specializations} @@ -298,13 +295,14 @@ The blob complex for a $1$-category $\cC$ on the circle is quasi-isomorphic to the Hochschild complex. \begin{equation*} -\xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).} +\xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).} \end{equation*} \end{thm:hochschild} Theorem \ref{thm:skein-modules} is immediate from the definition, and Theorem \ref{thm:hochschild} is established in \S \ref{sec:hochschild}. -We also note \S \ref{sec:comm_alg} which describes the blob complex when $\cC$ is a one of certain commutative algebras thought of as an $n$-category. +We also note \S \ref{sec:comm_alg} which describes the blob complex when $\cC$ is a one of +certain commutative algebras thought of as $n$-categories. \subsection{Structure of the blob complex} @@ -318,7 +316,7 @@ \label{thm:evaluation}% There is a chain map \begin{equation*} -\ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). +e_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). \end{equation*} such that \begin{enumerate} @@ -330,14 +328,15 @@ \begin{equation*} \xymatrix@C+2cm{ \CH{X} \otimes \bc_*(X) - \ar[r]_{\ev_{X}} \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y} & + \ar[r]_{e_{X}} \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y} & \bc_*(X) \ar[d]_{\gl_Y} \\ - \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{\ev_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) + \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{e_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) } \end{equation*} \end{enumerate} Moreover any such chain map is unique, up to an iterated homotopy. (That is, any pair of homotopies have a homotopy between them, and so on.) +\nn{revisit this after proof below has stabilized} \end{thm:CH} \newtheorem*{thm:CH-associativity}{Theorem \ref{thm:CH-associativity}} @@ -348,8 +347,8 @@ The chain map of Theorem \ref{thm:CH} is associative, in the sense that the following diagram commutes (up to homotopy). \begin{equation*} \xymatrix{ -\CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor \ev_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{\ev_X} \\ -\CH{X} \tensor \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) +\CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor e_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{e_X} \\ +\CH{X} \tensor \bc_*(X) \ar[r]^{e_X} & \bc_*(X) } \end{equation*} \end{thm:CH-associativity}