# HG changeset patch # User Kevin Walker # Date 1275696953 25200 # Node ID 1d76e832d32f98e4fcb841336d7d281c47a4ec4f # Parent 675f537354453d54ad4ce10bda7a4ed9c9a01a1a breaking long lines diff -r 675f53735445 -r 1d76e832d32f text/a_inf_blob.tex --- a/text/a_inf_blob.tex Fri Jun 04 17:00:18 2010 -0700 +++ b/text/a_inf_blob.tex Fri Jun 04 17:15:53 2010 -0700 @@ -16,7 +16,8 @@ \medskip An important technical tool in the proofs of this section is provided by the idea of `small blobs'. -Fix $\cU$, an open cover of $M$. Define the `small blob complex' $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ of all blob diagrams in which every blob is contained in some open set of $\cU$. +Fix $\cU$, an open cover of $M$. +Define the `small blob complex' $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ of all blob diagrams in which every blob is contained in some open set of $\cU$. \nn{KW: We need something a little stronger: Every blob diagram (even a 0-blob diagram) is splittable into pieces which are small w.r.t.\ $\cU$. If field have potentially large coupons/boxes, then this is a non-trivial constraint. On the other hand, we could probably get away with ignoring this point. @@ -46,11 +47,14 @@ \nn{need to settle on notation; proof and statement are inconsistent} \begin{thm} \label{product_thm} -Given a topological $n$-category $C$ and a $n{-}k$-manifold $F$, recall from Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $C^{\times F}$ defined by +Given a topological $n$-category $C$ and a $n{-}k$-manifold $F$, recall from +Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $C^{\times F}$ defined by \begin{equation*} C^{\times F}(B) = \cB_*(B \times F, C). \end{equation*} -Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the `old-fashioned' blob complex for $Y \times F$ with coefficients in $C$ and the `new-fangled' (i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $C^{\times F}$: +Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the `old-fashioned' +blob complex for $Y \times F$ with coefficients in $C$ and the `new-fangled' +(i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $C^{\times F}$: \begin{align*} \cB_*(Y \times F, C) & \htpy \cB_*(Y, C^{\times F}) \end{align*} @@ -305,7 +309,8 @@ Let $\cT$ denote the chain complex $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. Recall that this is a homotopy colimit based on decompositions of the interval $J$. -We define a map $\psi:\cT\to \bc_*(X)$. On filtration degree zero summands it is given +We define a map $\psi:\cT\to \bc_*(X)$. +On filtration degree zero summands it is given by gluing the pieces together to get a blob diagram on $X$. On filtration degree 1 and greater $\psi$ is zero. @@ -353,11 +358,18 @@ To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$. \begin{thm} \label{thm:map-recon} -The blob complex for $M$ with coefficients in the fundamental $A_\infty$ $n$-category for $T$ is quasi-isomorphic to singular chains on maps from $M$ to $T$. +The blob complex for $M$ with coefficients in the fundamental $A_\infty$ $n$-category for $T$ +is quasi-isomorphic to singular chains on maps from $M$ to $T$. $$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$ \end{thm} \begin{rem} -Lurie has shown in \cite[Theorem 3.8.6]{0911.0018} that the topological chiral homology of an $n$-manifold $M$ with coefficients in a certain $E_n$ algebra constructed from $T$ recovers the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n-1$-connected. This extra hypothesis is not surprising, in view of the idea described in Example \ref{ex:e-n-alg} that an $E_n$ algebra is roughly equivalent data to an $A_\infty$ $n$-category which is trivial at all but the topmost level. Ricardo Andrade also told us about a similar result. +Lurie has shown in \cite[Theorem 3.8.6]{0911.0018} that the topological chiral homology +of an $n$-manifold $M$ with coefficients in a certain $E_n$ algebra constructed from $T$ recovers +the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n-1$-connected. +This extra hypothesis is not surprising, in view of the idea described in Example \ref{ex:e-n-alg} +that an $E_n$ algebra is roughly equivalent data to an $A_\infty$ $n$-category which +is trivial at all but the topmost level. +Ricardo Andrade also told us about a similar result. \end{rem} \nn{proof is again similar to that of Theorem \ref{product_thm}. should probably say that explicitly} diff -r 675f53735445 -r 1d76e832d32f text/basic_properties.tex --- a/text/basic_properties.tex Fri Jun 04 17:00:18 2010 -0700 +++ b/text/basic_properties.tex Fri Jun 04 17:15:53 2010 -0700 @@ -3,9 +3,15 @@ \section{Basic properties of the blob complex} \label{sec:basic-properties} -In this section we complete the proofs of Properties 2-4. Throughout the paper, where possible, we prove results using Properties 1-4, rather than the actual definition of blob homology. This allows the possibility of future improvements to or alternatives on our definition. In fact, we hope that there may be a characterisation of blob homology in terms of Properties 1-4, but at this point we are unaware of one. +In this section we complete the proofs of Properties 2-4. +Throughout the paper, where possible, we prove results using Properties 1-4, +rather than the actual definition of blob homology. +This allows the possibility of future improvements to or alternatives on our definition. +In fact, we hope that there may be a characterisation of blob homology in +terms of Properties 1-4, but at this point we are unaware of one. -Recall Property \ref{property:disjoint-union}, that there is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. +Recall Property \ref{property:disjoint-union}, +that there is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. \begin{proof}[Proof of Property \ref{property:disjoint-union}] Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them @@ -15,7 +21,9 @@ In the other direction, any blob diagram on $X\du Y$ is equal (up to sign) to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines a pair of blob diagrams on $X$ and $Y$. -These two maps are compatible with our sign conventions. (We follow the usual convention for tensors products of complexes, as in e.g. \cite{MR1438306}: $d(a \tensor b) = da \tensor b + (-1)^{\deg(a)} a \tensor db$.) +These two maps are compatible with our sign conventions. +(We follow the usual convention for tensors products of complexes, +as in e.g. \cite{MR1438306}: $d(a \tensor b) = da \tensor b + (-1)^{\deg(a)} a \tensor db$.) The two maps are inverses of each other. \end{proof} @@ -43,7 +51,8 @@ Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$. \end{proof} -This proves Property \ref{property:contractibility} (the second half of the statement of this Property was immediate from the definitions). +This proves Property \ref{property:contractibility} (the second half of the +statement of this Property was immediate from the definitions). Note that even when there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$. @@ -92,7 +101,8 @@ Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$, we have the blob complex $\bc_*(X; a, b, c)$. If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on -$X$ to get blob diagrams on $X\sgl$. This proves Property \ref{property:gluing-map}, which we restate here in more detail. +$X$ to get blob diagrams on $X\sgl$. +This proves Property \ref{property:gluing-map}, which we restate here in more detail. \textbf{Property \ref{property:gluing-map}.}\emph{ There is a natural chain map diff -r 675f53735445 -r 1d76e832d32f text/blobdef.tex --- a/text/blobdef.tex Fri Jun 04 17:00:18 2010 -0700 +++ b/text/blobdef.tex Fri Jun 04 17:15:53 2010 -0700 @@ -57,9 +57,12 @@ (but keeping the blob label $u$). Note that the skein space $A(X)$ -is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. This is Property \ref{property:skein-modules}, and also used in the second half of Property \ref{property:contractibility}. +is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. +This is Property \ref{property:skein-modules}, and also used in the second +half of Property \ref{property:contractibility}. -Next, we want the vector space $\bc_2(X)$ to capture `the space of all relations (redundancies, syzygies) among the +Next, we want the vector space $\bc_2(X)$ to capture `the space of all relations +(redundancies, syzygies) among the local relations encoded in $\bc_1(X)$'. More specifically, a $2$-blob diagram, comes in one of two types, disjoint and nested. A disjoint 2-blob diagram consists of @@ -85,7 +88,8 @@ A nested 2-blob diagram consists of \begin{itemize} \item A pair of nested balls (blobs) $B_1 \sub B_2 \sub X$. -\item A field $r' \in \cC(B_2 \setminus B_1; c_1, c_2)$ (for some $c_1 \in \cC(\bdy B_1)$ and $c_2 \in \cC(\bdy B_2)$). +\item A field $r' \in \cC(B_2 \setminus B_1; c_1, c_2)$ +(for some $c_1 \in \cC(\bdy B_1)$ and $c_2 \in \cC(\bdy B_2)$). \item A field $r \in \cC(X \setminus B_2; c_2)$. \item A local relation field $u \in U(B_1; c_1)$. \end{itemize} @@ -114,7 +118,10 @@ \right) . \end{eqnarray*} For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign -(rather than a new, linearly independent 2-blob diagram). \nn{Hmm, I think we should be doing this for nested blobs too -- we shouldn't force the linear indexing of the blobs to have anything to do with the partial ordering by inclusion -- this is what happens below} +(rather than a new, linearly independent 2-blob diagram). +\nn{Hmm, I think we should be doing this for nested blobs too -- +we shouldn't force the linear indexing of the blobs to have anything to do with +the partial ordering by inclusion -- this is what happens below} Now for the general case. A $k$-blob diagram consists of @@ -158,7 +165,8 @@ \left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . \] Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. -The index $\overline{c}$ runs over all boundary conditions, again as described above and $j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are splittable along all of the blobs in $\overline{B}$. +The index $\overline{c}$ runs over all boundary conditions, again as described above and $j$ runs over all indices of twig blobs. +The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are splittable along all of the blobs in $\overline{B}$. The boundary map \[ @@ -180,7 +188,8 @@ The $(-1)^{j+1}$ factors imply that the terms of $\bd^2(b)$ all cancel. Thus we have a chain complex. -Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, is immediately obvious from the definition. A homeomorphism acts in an obvious on blobs and on fields. +Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, is immediately obvious from the definition. +A homeomorphism acts in an obvious on blobs and on fields. We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, to be the union of the blobs of $b$. @@ -195,8 +204,10 @@ (equivalently, to each rooted tree) according to the following rules: \begin{itemize} \item $p(\emptyset) = pt$, where $\emptyset$ denotes a 0-blob diagram or empty tree; -\item $p(a \du b) = p(a) \times p(b)$, where $a \du b$ denotes the distant (non-overlapping) union of two blob diagrams (equivalently, join two trees at the roots); and -\item $p(\bar{b}) = \kone(p(b))$, where $\bar{b}$ is obtained from $b$ by adding an outer blob which encloses all the others (equivalently, add a new edge to the root, with the new vertex becoming the root). +\item $p(a \du b) = p(a) \times p(b)$, where $a \du b$ denotes the distant (non-overlapping) union +of two blob diagrams (equivalently, join two trees at the roots); and +\item $p(\bar{b}) = \kone(p(b))$, where $\bar{b}$ is obtained from $b$ by adding an outer blob which +encloses all the others (equivalently, add a new edge to the root, with the new vertex becoming the root). \end{itemize} For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while a diagram of $k$ disjoint blobs corresponds to a $k$-cube. diff -r 675f53735445 -r 1d76e832d32f text/comm_alg.tex --- a/text/comm_alg.tex Fri Jun 04 17:00:18 2010 -0700 +++ b/text/comm_alg.tex Fri Jun 04 17:15:53 2010 -0700 @@ -13,7 +13,10 @@ The goal of this \nn{subsection?} is to compute $\bc_*(M^n, C)$ for various commutative algebras $C$. -Moreover, we conjecture that the blob complex $\bc_*(M^n, $C$)$, for $C$ a commutative algebra is homotopy equivalent to the higher Hochschild complex for $M^n$ with coefficients in $C$ (see \cite{MR0339132, MR1755114, MR2383113}). This possibility was suggested to us by Thomas Tradler. +Moreover, we conjecture that the blob complex $\bc_*(M^n, $C$)$, for $C$ a commutative +algebra is homotopy equivalent to the higher Hochschild complex for $M^n$ with +coefficients in $C$ (see \cite{MR0339132, MR1755114, MR2383113}). +This possibility was suggested to us by Thomas Tradler. \medskip @@ -108,8 +111,12 @@ \nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} Let us check this directly. -The algebra $k[t]$ has Koszul resolution $k[t] \tensor k[t] \xrightarrow{t\tensor 1 - 1 \tensor t} k[t] \tensor k[t]$, which has coinvariants $k[t] \xrightarrow{0} k[t]$. This is equal to its homology, so we have $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and zero for $i\ge 2$. -(See also \cite[3.2.2]{MR1600246}.) This computation also tells us the $t$-gradings: $HH_0(k[t]) \iso k[t]$ is in the usual grading, and $HH_1(k[t]) \iso k[t]$ is shifted up by one. +The algebra $k[t]$ has Koszul resolution +$k[t] \tensor k[t] \xrightarrow{t\tensor 1 - 1 \tensor t} k[t] \tensor k[t]$, +which has coinvariants $k[t] \xrightarrow{0} k[t]$. +This is equal to its homology, so we have $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and zero for $i\ge 2$. +(See also \cite[3.2.2]{MR1600246}.) This computation also tells us the $t$-gradings: +$HH_0(k[t]) \iso k[t]$ is in the usual grading, and $HH_1(k[t]) \iso k[t]$ is shifted up by one. We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other. The fixed points of this flow are the equally spaced configurations. @@ -152,7 +159,8 @@ \] Let us check that this is also the singular homology of $\Sigma_m^\infty(S^1)$. We will content ourselves with the case $k = \z$. -One can define a flow on $\Sigma_m^\infty(S^1)$ where points of the same color repel each other and points of different colors do not interact. +One can define a flow on $\Sigma_m^\infty(S^1)$ where points of the +same color repel each other and points of different colors do not interact. This shows that a component $X$ of $\Sigma_m^\infty(S^1)$ is homotopy equivalent to the torus $(S^1)^l$, where $l$ is the number of non-zero entries in the $m$-tuple corresponding to $X$. diff -r 675f53735445 -r 1d76e832d32f text/deligne.tex --- a/text/deligne.tex Fri Jun 04 17:00:18 2010 -0700 +++ b/text/deligne.tex Fri Jun 04 17:15:53 2010 -0700 @@ -11,7 +11,8 @@ (Proposition \ref{prop:deligne} below). Then we sketch the proof. -\nn{Does this generalisation encompass Kontsevich's proposed generalisation from \cite[\S2.5]{MR1718044}, that (I think...) the Hochschild homology of an $E_n$ algebra is an $E_{n+1}$ algebra? -S} +\nn{Does this generalisation encompass Kontsevich's proposed generalisation from \cite[\S2.5]{MR1718044}, +that (I think...) the Hochschild homology of an $E_n$ algebra is an $E_{n+1}$ algebra? -S} %from http://www.ams.org/mathscinet-getitem?mr=1805894 %Different versions of the geometric counterpart of Deligne's conjecture have been proven by Tamarkin [``Formality of chain operad of small squares'', preprint, http://arXiv.org/abs/math.QA/9809164], the reviewer [in Confˇrence Moshˇ Flato 1999, Vol. II (Dijon), 307--331, Kluwer Acad. Publ., Dordrecht, 2000; MR1805923 (2002d:55009)], and J. E. McClure and J. H. Smith [``A solution of Deligne's conjecture'', preprint, http://arXiv.org/abs/math.QA/9910126] (see also a later simplified version [J. E. McClure and J. H. Smith, ``Multivariable cochain operations and little $n$-cubes'', preprint, http://arXiv.org/abs/math.QA/0106024]). The paper under review gives another proof of Deligne's conjecture, which, as the authors indicate, may be generalized to a proof of a higher-dimensional generalization of Deligne's conjecture, suggested in [M. Kontsevich, Lett. Math. Phys. 48 (1999), no. 1, 35--72; MR1718044 (2000j:53119)]. diff -r 675f53735445 -r 1d76e832d32f text/hochschild.tex --- a/text/hochschild.tex Fri Jun 04 17:00:18 2010 -0700 +++ b/text/hochschild.tex Fri Jun 04 17:15:53 2010 -0700 @@ -7,7 +7,11 @@ greater than zero. In this section we analyze the blob complex in dimension $n=1$. We find that $\bc_*(S^1, \cC)$ is homotopy equivalent to the -Hochschild complex of the 1-category $\cC$. (Recall from \S \ref{sec:example:traditional-n-categories(fields)} that a $1$-category gives rise to a $1$-dimensional system of fields; as usual, talking about the blob complex with coefficients in a $n$-category means first passing to the corresponding $n$ dimensional system of fields.) +Hochschild complex of the 1-category $\cC$. +(Recall from \S \ref{sec:example:traditional-n-categories(fields)} that a +$1$-category gives rise to a $1$-dimensional system of fields; as usual, +talking about the blob complex with coefficients in a $n$-category means +first passing to the corresponding $n$ dimensional system of fields.) Thus the blob complex is a natural generalization of something already known to be interesting in higher homological degrees. @@ -67,12 +71,14 @@ usual Hochschild complex for $C$. \end{thm} -This follows from two results. First, we see that +This follows from two results. +First, we see that \begin{lem} \label{lem:module-blob}% The complex $K_*(C)$ (here $C$ is being thought of as a $C$-$C$-bimodule, not a category) is homotopy equivalent to the blob complex -$\bc_*(S^1; C)$. (Proof later.) +$\bc_*(S^1; C)$. +(Proof later.) \end{lem} Next, we show that for any $C$-$C$-bimodule $M$, @@ -114,17 +120,19 @@ $$\cP_*(M) \iso \coinv(F_*).$$ % Observe that there's a quotient map $\pi: F_0 \onto M$, and by -construction the cone of the chain map $\pi: F_* \to M$ is acyclic. Now -construct the total complex $\cP_i(F_j)$, with $i,j \geq 0$, graded by -$i+j$. We have two chain maps +construction the cone of the chain map $\pi: F_* \to M$ is acyclic. +Now construct the total complex $\cP_i(F_j)$, with $i,j \geq 0$, graded by $i+j$. +We have two chain maps \begin{align*} \cP_i(F_*) & \xrightarrow{\cP_i(\pi)} \cP_i(M) \\ \intertext{and} \cP_*(F_j) & \xrightarrow{\cP_0(F_j) \onto H_0(\cP_*(F_j))} \coinv(F_j). \end{align*} -The cone of each chain map is acyclic. In the first case, this is because the `rows' indexed by $i$ are acyclic since $\HC_i$ is exact. +The cone of each chain map is acyclic. +In the first case, this is because the `rows' indexed by $i$ are acyclic since $\HC_i$ is exact. In the second case, this is because the `columns' indexed by $j$ are acyclic, since $F_j$ is free. -Because the cones are acyclic, the chain maps are quasi-isomorphisms. Composing one with the inverse of the other, we obtain the desired quasi-isomorphism +Because the cones are acyclic, the chain maps are quasi-isomorphisms. +Composing one with the inverse of the other, we obtain the desired quasi-isomorphism $$\cP_*(M) \quismto \coinv(F_*).$$ %If $M$ is free, that is, a direct sum of copies of @@ -150,7 +158,8 @@ %and higher homology groups are determined by lower ones in $\HC_*(K)$, and %hence recursively as coinvariants of some other bimodule. -Proposition \ref{prop:hoch} then follows from the following lemmas, establishing that $K_*$ has precisely these required properties. +Proposition \ref{prop:hoch} then follows from the following lemmas, +establishing that $K_*$ has precisely these required properties. \begin{lem} \label{lem:hochschild-additive}% Directly from the definition, $K_*(M_1 \oplus M_2) \cong K_*(M_1) \oplus K_*(M_2)$. @@ -185,7 +194,8 @@ We want to define a homotopy inverse to the above inclusion, but before doing so we must replace $\bc_*(S^1)$ with a homotopy equivalent subcomplex. Let $J_* \sub \bc_*(S^1)$ be the subcomplex where * does not lie on the boundary -of any blob. Note that the image of $i$ is contained in $J_*$. +of any blob. +Note that the image of $i$ is contained in $J_*$. Note also that in $\bc_*(S^1)$ (away from $J_*$) a blob diagram could have multiple (nested) blobs whose boundaries contain *, on both the right and left of *. @@ -219,10 +229,13 @@ every blob in the diagram. Note that for any chain $x \in \bc_*(S^1)$, $x \in L_*^\ep$ for sufficiently small $\ep$. -We define a degree $1$ map $j_\ep: L_*^\ep \to L_*^\ep$ as follows. Let $x \in L_*^\ep$ be a blob diagram. +We define a degree $1$ map $j_\ep: L_*^\ep \to L_*^\ep$ as follows. +Let $x \in L_*^\ep$ be a blob diagram. \nn{maybe add figures illustrating $j_\ep$?} -If $*$ is not contained in any twig blob, we define $j_\ep(x)$ by adding $N_\ep$ as a new twig blob, with label $y - s(y)$ where $y$ is the restriction -of $x$ to $N_\ep$. If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$, +If $*$ is not contained in any twig blob, we define $j_\ep(x)$ by adding +$N_\ep$ as a new twig blob, with label $y - s(y)$ where $y$ is the restriction +of $x$ to $N_\ep$. +If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$, write $y_i$ for the restriction of $z_i$ to $N_\ep$, and let $x_i$ be equal to $x$ on $S^1 \setmin B$, equal to $z_i$ on $B \setmin N_\ep$, and have an additional blob $N_\ep$ with label $y_i - s(y_i)$. @@ -256,14 +269,24 @@ \] and similarly for $\hat{g}$. Most of what we need to check is easy. -Suppose we have $\sum_i (a_i \tensor k_i \tensor b_i) \in \ker(C \tensor K \tensor C \to K)$, assuming without loss of generality that $\{a_i \tensor b_i\}_i$ is linearly independent in $C \tensor C$, and $\hat{f}(a \tensor k \tensor b) = 0 \in \ker(C \tensor E \tensor C \to E)$. We must then have $f(k_i) = 0 \in E$ for each $i$, which implies $k_i=0$ itself. -If $\sum_i (a_i \tensor e_i \tensor b_i) \in \ker(C \tensor E \tensor C \to E)$ is in the image of $\ker(C \tensor K \tensor C \to K)$ under $\hat{f}$, again by assuming the set $\{a_i \tensor b_i\}_i$ is linearly independent we can deduce that each -$e_i$ is in the image of the original $f$, and so is in the kernel of the original $g$, and so $\hat{g}(\sum_i a_i \tensor e_i \tensor b_i) = 0$. -If $\hat{g}(\sum_i a_i \tensor e_i \tensor b_i) = 0$, then each $g(e_i) = 0$, so $e_i = f(\widetilde{e_i})$ for some $\widetilde{e_i} \in K$, and $\sum_i a_i \tensor e_i \tensor b_i = \hat{f}(\sum_i a_i \tensor \widetilde{e_i} \tensor b_i)$. -Finally, the interesting step is in checking that any $q = \sum_i a_i \tensor q_i \tensor b_i$ such that $\sum_i a_i q_i b_i = 0$ is in the image of $\ker(C \tensor E \tensor C \to C)$ under $\hat{g}$. -For each $i$, we can find $\widetilde{q_i}$ so $g(\widetilde{q_i}) = q_i$. However $\sum_i a_i \widetilde{q_i} b_i$ need not be zero. +Suppose we have $\sum_i (a_i \tensor k_i \tensor b_i) \in \ker(C \tensor K \tensor C \to K)$, +assuming without loss of generality that $\{a_i \tensor b_i\}_i$ is linearly independent in $C \tensor C$, +and $\hat{f}(a \tensor k \tensor b) = 0 \in \ker(C \tensor E \tensor C \to E)$. +We must then have $f(k_i) = 0 \in E$ for each $i$, which implies $k_i=0$ itself. +If $\sum_i (a_i \tensor e_i \tensor b_i) \in \ker(C \tensor E \tensor C \to E)$ +is in the image of $\ker(C \tensor K \tensor C \to K)$ under $\hat{f}$, +again by assuming the set $\{a_i \tensor b_i\}_i$ is linearly independent we can deduce that each +$e_i$ is in the image of the original $f$, and so is in the kernel of the original $g$, +and so $\hat{g}(\sum_i a_i \tensor e_i \tensor b_i) = 0$. +If $\hat{g}(\sum_i a_i \tensor e_i \tensor b_i) = 0$, then each $g(e_i) = 0$, so $e_i = f(\widetilde{e_i})$ +for some $\widetilde{e_i} \in K$, and $\sum_i a_i \tensor e_i \tensor b_i = \hat{f}(\sum_i a_i \tensor \widetilde{e_i} \tensor b_i)$. +Finally, the interesting step is in checking that any $q = \sum_i a_i \tensor q_i \tensor b_i$ +such that $\sum_i a_i q_i b_i = 0$ is in the image of $\ker(C \tensor E \tensor C \to C)$ under $\hat{g}$. +For each $i$, we can find $\widetilde{q_i}$ so $g(\widetilde{q_i}) = q_i$. +However $\sum_i a_i \widetilde{q_i} b_i$ need not be zero. Consider then $$\widetilde{q} = \sum_i (a_i \tensor \widetilde{q_i} \tensor b_i) - 1 \tensor (\sum_i a_i \widetilde{q_i} b_i) \tensor 1.$$ Certainly -$\widetilde{q} \in \ker(C \tensor E \tensor C \to E)$. Further, +$\widetilde{q} \in \ker(C \tensor E \tensor C \to E)$. +Further, \begin{align*} \hat{g}(\widetilde{q}) & = \sum_i (a_i \tensor g(\widetilde{q_i}) \tensor b_i) - 1 \tensor (\sum_i a_i g(\widetilde{q_i}) b_i) \tensor 1 \\ & = q - 0 @@ -275,32 +298,44 @@ \label{eq:ker-functor}% M \mapsto \ker(C^{\tensor k} \tensor M \tensor C^{\tensor l} \to M) \end{equation} -are all exact too. Moreover, tensor products of such functors with each +are all exact too. +Moreover, tensor products of such functors with each other and with $C$ or $\ker(C^{\tensor k} \to C)$ (e.g., producing the functor $M \mapsto \ker(M \tensor C \to M) \tensor C \tensor \ker(C \tensor C \to M)$) are all still exact. Finally, then we see that the functor $K_*$ is simply an (infinite) -direct sum of copies of this sort of functor. The direct sum is indexed by -configurations of nested blobs and of labels; for each such configuration, we have one of the above tensor product functors, -with the labels of twig blobs corresponding to tensor factors as in \eqref{eq:ker-functor} or $\ker(C^{\tensor k} \to C)$ (depending on whether they contain a marked point $p_i$), and all other labelled points corresponding +direct sum of copies of this sort of functor. +The direct sum is indexed by +configurations of nested blobs and of labels; for each such configuration, we have one of +the above tensor product functors, +with the labels of twig blobs corresponding to tensor factors as in \eqref{eq:ker-functor} +or $\ker(C^{\tensor k} \to C)$ (depending on whether they contain a marked point $p_i$), and all other labelled points corresponding to tensor factors of $C$ and $M$. \end{proof} \begin{proof}[Proof of Lemma \ref{lem:hochschild-coinvariants}] We show that $H_0(K_*(M))$ is isomorphic to the coinvariants of $M$. -We define a map $\ev: K_0(M) \to M$. If $x \in K_0(M)$ has the label $m \in M$ at $*$, and labels $c_i \in C$ at the other labeled points of $S^1$, reading clockwise from $*$, -we set $\ev(x) = m c_1 \cdots c_k$. We can think of this as $\ev : M \tensor C^{\tensor k} \to M$, for each direct summand of $K_0(M)$ indexed by a configuration of labeled points. +We define a map $\ev: K_0(M) \to M$. +If $x \in K_0(M)$ has the label $m \in M$ at $*$, and labels $c_i \in C$ at the other +labeled points of $S^1$, reading clockwise from $*$, +we set $\ev(x) = m c_1 \cdots c_k$. +We can think of this as $\ev : M \tensor C^{\tensor k} \to M$, for each direct summand of +$K_0(M)$ indexed by a configuration of labeled points. There is a quotient map $\pi: M \to \coinv{M}$. We claim that the composition $\pi \compose \ev$ is well-defined on the quotient $H_0(K_*(M))$; i.e.\ that $\pi(\ev(\bd y)) = 0$ for all $y \in K_1(M)$. There are two cases, depending on whether the blob of $y$ contains the point *. If it doesn't, then -suppose $y$ has label $m$ at $*$, labels $c_i$ at other labeled points outside the blob, and the field inside the blob is a sum, with the $j$-th term having -labeled points $d_{j,i}$. Then $\sum_j d_{j,1} \tensor \cdots \tensor d_{j,k_j} \in \ker(\DirectSum_k C^{\tensor k} \to C)$, and so +suppose $y$ has label $m$ at $*$, labels $c_i$ at other labeled points outside the blob, +and the field inside the blob is a sum, with the $j$-th term having +labeled points $d_{j,i}$. +Then $\sum_j d_{j,1} \tensor \cdots \tensor d_{j,k_j} \in \ker(\DirectSum_k C^{\tensor k} \to C)$, and so $\ev(\bdy y) = 0$, because $$C^{\tensor \ell_1} \tensor \ker(\DirectSum_k C^{\tensor k} \to C) \tensor C^{\tensor \ell_2} \subset \ker(\DirectSum_k C^{\tensor k} \to C).$$ -Similarly, if $*$ is contained in the blob, then the blob label is a sum, with the $j$-th term have labelled points $d_{j,i}$ to the left of $*$, $m_j$ at $*$, and $d_{j,i}'$ to the right of $*$, -and there are labels $c_i$ at the labeled points outside the blob. We know that +Similarly, if $*$ is contained in the blob, then the blob label is a sum, with the +$j$-th term have labelled points $d_{j,i}$ to the left of $*$, $m_j$ at $*$, and $d_{j,i}'$ to the right of $*$, +and there are labels $c_i$ at the labeled points outside the blob. +We know that $$\sum_j d_{j,1} \tensor \cdots \tensor d_{j,k_j} \tensor m_j \tensor d_{j,1}' \tensor \cdots \tensor d_{j,k'_j}' \in \ker(\DirectSum_{k,k'} C^{\tensor k} \tensor M \tensor C^{\tensor k'} \tensor \to M),$$ and so \begin{align*} @@ -310,7 +345,8 @@ \end{align*} where this time we use the fact that we're mapping to $\coinv{M}$, not just $M$. -The map $\pi \compose \ev: H_0(K_*(M)) \to \coinv{M}$ is clearly surjective ($\ev$ surjects onto $M$); we now show that it's injective. +The map $\pi \compose \ev: H_0(K_*(M)) \to \coinv{M}$ is clearly +surjective ($\ev$ surjects onto $M$); we now show that it's injective. This is equivalent to showing that \[ \ev\inv(\ker(\pi)) \sub \bd K_1(M) . @@ -340,7 +376,8 @@ \begin{proof}[Proof of Lemma \ref{lem:hochschild-free}] We show that $K_*(C\otimes C)$ is -quasi-isomorphic to the 0-step complex $C$. We'll do this in steps, establishing quasi-isomorphisms and homotopy equivalences +quasi-isomorphic to the 0-step complex $C$. +We'll do this in steps, establishing quasi-isomorphisms and homotopy equivalences $$K_*(C \tensor C) \quismto K'_* \htpyto K''_* \quismto C.$$ Let $K'_* \sub K_*(C\otimes C)$ be the subcomplex where the label of @@ -355,7 +392,8 @@ %and the two boundary points of $N_\ep$ are not labeled points of $b$. For a field $y$ on $N_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$ labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$. -(See Figure \ref{fig:sy}.) Note that $y - s_\ep(y) \in U(N_\ep)$. +(See Figure \ref{fig:sy}.) +Note that $y - s_\ep(y) \in U(N_\ep)$. Let $\sigma_\ep: K_*^\ep \to K_*^\ep$ be the chain map given by replacing the restriction $y$ to $N_\ep$ of each field appearing in an element of $K_*^\ep$ with $s_\ep(y)$. @@ -512,7 +550,8 @@ \begin{equation*} \mathfig{0.4}{hochschild/2-chains-1} \qquad \mathfig{0.4}{hochschild/2-chains-2} \end{equation*} -\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.} +\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} @@ -529,7 +568,8 @@ \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. +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. diff -r 675f53735445 -r 1d76e832d32f text/ncat.tex --- a/text/ncat.tex Fri Jun 04 17:00:18 2010 -0700 +++ b/text/ncat.tex Fri Jun 04 17:15:53 2010 -0700 @@ -316,7 +316,6 @@ \[ (a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') . \] -\nn{if pinched boundary, then remove first case above} \item Product morphisms are associative: \[