--- 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}
--- 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
--- 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.
--- 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 $X$.
--- 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)].
--- 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.
--- 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:
\[