author | Kevin Walker <kevin@canyon23.net> |
Mon, 23 Aug 2010 21:19:55 -0700 | |
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%!TEX root = ../blob1.tex |
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\section{Hochschild homology when \texorpdfstring{$n=1$}{n=1}} |
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\label{sec:hochschild} |
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So far we have provided no evidence that blob homology is interesting in degrees |
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greater than zero. |
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In this section we analyze the blob complex in dimension $n=1$. |
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We find that $\bc_*(S^1, \cC)$ is homotopy equivalent to the |
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Hochschild complex of the 1-category $\cC$. |
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(Recall from \S \ref{sec:example:traditional-n-categories(fields)} that a |
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$1$-category gives rise to a $1$-dimensional system of fields; as usual, |
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talking about the blob complex with coefficients in a $n$-category means |
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first passing to the corresponding $n$ dimensional system of fields.) |
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Thus the blob complex is a natural generalization of something already |
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known to be interesting in higher homological degrees. |
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It is also worth noting that the original idea for the blob complex came from trying |
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to find a more ``local" description of the Hochschild complex. |
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Let $C$ be a *-1-category. |
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Then specializing the definition of the associated system of fields from \S \ref{sec:example:traditional-n-categories(fields)} above to the case $n=1$ we have: |
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\begin{itemize} |
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\item $\cC(pt) = \ob(C)$ . |
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\item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$. |
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Then an element of $\cC(R; c)$ is a collection of (transversely oriented) |
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points in the interior |
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of $R$, each labeled by a morphism of $C$. |
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The intervals between the points are labeled by objects of $C$, consistent with |
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the boundary condition $c$ and the domains and ranges of the point labels. |
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\item There is an evaluation map $e: \cC(I; a, b) \to \mor(a, b)$ given by |
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composing the morphism labels of the points. |
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Note that we also need the * of *-1-category here in order to make all the morphisms point |
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the same way. |
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\item For $x \in \mor(a, b)$ let $\chi(x) \in \cC(I; a, b)$ be the field with a single |
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point (at some standard location) labeled by $x$. |
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Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the |
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form $y - \chi(e(y))$. |
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Thus we can, if we choose, restrict the blob twig labels to things of this form. |
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\end{itemize} |
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We want to show that $\bc_*(S^1)$ is homotopy equivalent to the |
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Hochschild complex of $C$. |
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In order to prove this we will need to extend the |
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definition of the blob complex to allow points to also |
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be labeled by elements of $C$-$C$-bimodules. |
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(See Subsections \ref{moddecss} and \ref{ssec:spherecat} for a more general version of this construction that applies in all dimensions.) |
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Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$. |
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We define a blob-like complex $K_*(S^1, (p_i), (M_i))$. |
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The fields have elements of $M_i$ labeling |
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the fixed points $p_i$ and elements of $C$ labeling other (variable) points. |
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As before, the regions between the marked points are labeled by |
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objects of $\cC$. |
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The blob twig labels lie in kernels of evaluation maps. |
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(The range of these evaluation maps is a tensor product (over $C$) of $M_i$'s, |
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corresponding to the $p_i$'s that lie within the twig blob.) |
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Let $K_*(M) = K_*(S^1, (*), (M))$, where $* \in S^1$ is some standard base point. |
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In other words, fields for $K_*(M)$ have an element of $M$ at the fixed point $*$ |
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and elements of $C$ at variable other points. |
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In the theorems, propositions and lemmas below we make various claims |
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about complexes being homotopy equivalent. |
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In all cases the complexes in question are free (and hence projective), |
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so it suffices to show that they are quasi-isomorphic. |
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We claim that |
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\begin{thm} |
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\label{thm:hochschild} |
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The blob complex $\bc_*(S^1; C)$ on the circle is homotopy equivalent to the |
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usual Hochschild complex for $C$. |
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\end{thm} |
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This follows from two results. |
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First, we see that |
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\begin{lem} |
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\label{lem:module-blob}% |
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The complex $K_*(C)$ (here $C$ is being thought of as a |
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$C$-$C$-bimodule, not a category) is homotopy equivalent to the blob complex |
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$\bc_*(S^1; C)$. |
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\end{lem} |
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The proof appears below. |
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Next, we show that for any $C$-$C$-bimodule $M$, |
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\begin{prop} \label{prop:hoch} |
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The complex $K_*(M)$ is homotopy equivalent to $\HC_*(M)$, the usual |
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Hochschild complex of $M$. |
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\end{prop} |
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\begin{proof} |
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Recall that the usual Hochschild complex of $M$ is uniquely determined, |
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up to quasi-isomorphism, by the following properties: |
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\begin{enumerate} |
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\item \label{item:hochschild-additive}% |
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$\HC_*(M_1 \oplus M_2) \cong \HC_*(M_1) \oplus \HC_*(M_2)$. |
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\item \label{item:hochschild-exact}% |
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An exact sequence $0 \to M_1 \into M_2 \onto M_3 \to 0$ gives rise to an |
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exact sequence $0 \to \HC_*(M_1) \into \HC_*(M_2) \onto \HC_*(M_3) \to 0$. |
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\item \label{item:hochschild-coinvariants}% |
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$\HH_0(M)$ is isomorphic to the coinvariants of $M$, $\coinv(M) = |
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M/\langle cm-mc \rangle$. |
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\item \label{item:hochschild-free}% |
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$\HC_*(C\otimes C)$ is contractible. |
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(Here $C\otimes C$ denotes |
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the free $C$-$C$-bimodule with one generator.) |
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That is, $\HC_*(C\otimes C)$ is |
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quasi-isomorphic to its $0$-th homology (which in turn, by \ref{item:hochschild-coinvariants} |
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above, is just $C$) via the quotient map $\HC_0 \onto \HH_0$. |
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\end{enumerate} |
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(Together, these just say that Hochschild homology is ``the derived functor of coinvariants".) |
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We'll first recall why these properties are characteristic. |
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Take some $C$-$C$ bimodule $M$, and choose a free resolution |
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\begin{equation*} |
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\cdots \to F_2 \xrightarrow{f_2} F_1 \xrightarrow{f_1} F_0. |
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\end{equation*} |
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We will show that for any functor $\cP$ satisfying properties |
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\ref{item:hochschild-additive}, \ref{item:hochschild-exact}, |
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\ref{item:hochschild-coinvariants} and \ref{item:hochschild-free}, there |
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is a quasi-isomorphism |
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$$\cP_*(M) \iso \coinv(F_*).$$ |
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% |
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Observe that there's a quotient map $\pi: F_0 \onto M$, and by |
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construction the cone of the chain map $\pi: F_* \to M$ is acyclic. |
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Now construct the total complex $\cP_i(F_j)$, with $i,j \geq 0$, graded by $i+j$. |
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We have two chain maps |
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\begin{align*} |
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\cP_i(F_*) & \xrightarrow{\cP_i(\pi)} \cP_i(M) \\ |
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\intertext{and} |
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\cP_*(F_j) & \xrightarrow{\cP_0(F_j) \onto H_0(\cP_*(F_j))} \coinv(F_j). |
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\end{align*} |
342 | 131 |
The cone of each chain map is acyclic. |
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In the first case, this is because the ``rows" indexed by $i$ are acyclic since $\cP_i$ is exact. |
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In the second case, this is because the ``columns" indexed by $j$ are acyclic, since $F_j$ is free. |
342 | 134 |
Because the cones are acyclic, the chain maps are quasi-isomorphisms. |
135 |
Composing one with the inverse of the other, we obtain the desired quasi-isomorphism |
|
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$$\cP_*(M) \quismto \coinv(F_*).$$ |
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137 |
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138 |
%If $M$ is free, that is, a direct sum of copies of |
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%$C \tensor C$, then properties \ref{item:hochschild-additive} and |
136 | 140 |
%\ref{item:hochschild-free} determine $\HC_*(M)$. Otherwise, choose some |
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%free cover $F \onto M$, and define $K$ to be this map's kernel. Thus we |
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%have a short exact sequence $0 \to K \into F \onto M \to 0$, and hence a |
136 | 143 |
%short exact sequence of complexes $0 \to \HC_*(K) \into \HC_*(F) \onto \HC_*(M) |
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%\to 0$. Such a sequence gives a long exact sequence on homology |
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145 |
%\begin{equation*} |
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%%\begin{split} |
136 | 147 |
%\cdots \to \HH_{i+1}(F) \to \HH_{i+1}(M) \to \HH_i(K) \to \HH_i(F) \to \cdots % \\ |
148 |
%%\cdots \to \HH_1(F) \to \HH_1(M) \to \HH_0(K) \to \HH_0(F) \to \HH_0(M). |
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%%\end{split} |
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%\end{equation*} |
136 | 151 |
%For any $i \geq 1$, $\HH_{i+1}(F) = \HH_i(F) = 0$, by properties |
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%\ref{item:hochschild-additive} and \ref{item:hochschild-free}, and so |
136 | 153 |
%$\HH_{i+1}(M) \iso \HH_i(F)$. For $i=0$, \todo{}. |
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154 |
% |
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155 |
%This tells us how to |
136 | 156 |
%compute every homology group of $\HC_*(M)$; we already know $\HH_0(M)$ |
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157 |
%(it's just coinvariants, by property \ref{item:hochschild-coinvariants}), |
136 | 158 |
%and higher homology groups are determined by lower ones in $\HC_*(K)$, and |
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%hence recursively as coinvariants of some other bimodule. |
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160 |
|
342 | 161 |
Proposition \ref{prop:hoch} then follows from the following lemmas, |
162 |
establishing that $K_*$ has precisely these required properties. |
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163 |
\begin{lem} |
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\label{lem:hochschild-additive}% |
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Directly from the definition, $K_*(M_1 \oplus M_2) \cong K_*(M_1) \oplus K_*(M_2)$. |
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166 |
\end{lem} |
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167 |
\begin{lem} |
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168 |
\label{lem:hochschild-exact}% |
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169 |
An exact sequence $0 \to M_1 \into M_2 \onto M_3 \to 0$ gives rise to an |
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170 |
exact sequence $0 \to K_*(M_1) \into K_*(M_2) \onto K_*(M_3) \to 0$. |
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171 |
\end{lem} |
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172 |
\begin{lem} |
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173 |
\label{lem:hochschild-coinvariants}% |
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174 |
$H_0(K_*(M))$ is isomorphic to the coinvariants of $M$. |
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175 |
\end{lem} |
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176 |
\begin{lem} |
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177 |
\label{lem:hochschild-free}% |
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$K_*(C\otimes C)$ is quasi-isomorphic to $H_0(K_*(C \otimes C)) \iso C$. |
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179 |
\end{lem} |
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180 |
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181 |
The remainder of this section is devoted to proving Lemmas |
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182 |
\ref{lem:module-blob}, |
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183 |
\ref{lem:hochschild-exact}, \ref{lem:hochschild-coinvariants} and |
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184 |
\ref{lem:hochschild-free}. |
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185 |
\end{proof} |
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186 |
|
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187 |
\subsection{Technical details} |
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188 |
\begin{proof}[Proof of Lemma \ref{lem:module-blob}] |
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189 |
We show that $K_*(C)$ is quasi-isomorphic to $\bc_*(S^1)$. |
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190 |
$K_*(C)$ differs from $\bc_*(S^1)$ only in that the base point * |
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191 |
is always a labeled point in $K_*(C)$, while in $\bc_*(S^1)$ it may or may not be. |
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192 |
In particular, there is an inclusion map $i: K_*(C) \to \bc_*(S^1)$. |
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193 |
|
219 | 194 |
We want to define a homotopy inverse to the above inclusion, but before doing so |
195 |
we must replace $\bc_*(S^1)$ with a homotopy equivalent subcomplex. |
|
221 | 196 |
Let $J_* \sub \bc_*(S^1)$ be the subcomplex where * does not lie on the boundary |
342 | 197 |
of any blob. |
198 |
Note that the image of $i$ is contained in $J_*$. |
|
219 | 199 |
Note also that in $\bc_*(S^1)$ (away from $J_*$) |
200 |
a blob diagram could have multiple (nested) blobs whose |
|
201 |
boundaries contain *, on both the right and left of *. |
|
202 |
||
203 |
We claim that $J_*$ is homotopy equivalent to $\bc_*(S^1)$. |
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220 | 204 |
Let $F_*^\ep \sub \bc_*(S^1)$ be the subcomplex where either |
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205 |
(a) the point * is not on the boundary of any blob or |
409 | 206 |
(b) there are no labeled points or blob boundaries within distance $\ep$ of *, |
207 |
other than blob boundaries at * itself. |
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220 | 208 |
Note that all blob diagrams are in $F_*^\ep$ for $\ep$ sufficiently small. |
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209 |
Let $b$ be a blob diagram in $F_*^\ep$. |
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210 |
Define $f(b)$ to be the result of moving any blob boundary points which lie on * |
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211 |
to distance $\ep$ from *. |
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212 |
(Move right or left so as to shrink the blob.) |
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213 |
Extend to get a chain map $f: F_*^\ep \to F_*^\ep$. |
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By Lemma \ref{support-shrink}, $f$ is homotopic to the identity. |
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215 |
Since the image of $f$ is in $J_*$, and since any blob chain is in $F_*^\ep$ |
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216 |
for $\ep$ sufficiently small, we have that $J_*$ is homotopic to all of $\bc_*(S^1)$. |
220 | 217 |
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218 |
We now define a homotopy inverse $s: J_* \to K_*(C)$ to the inclusion $i$. |
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If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
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* is a labeled point in $y$. |
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Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
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222 |
Extending linearly, we get the desired map $s: J_* \to K_*(C)$. |
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223 |
It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
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224 |
|
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Let $N_\ep$ denote the ball of radius $\ep$ around *. |
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Let $L_*^\ep \sub J_*$ be the subcomplex |
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spanned by blob diagrams |
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where there are no labeled points |
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in $N_\ep$, except perhaps $*$, and $N_\ep$ is either disjoint from or contained in |
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every blob in the diagram. |
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Note that for any chain $x \in J_*$, $x \in L_*^\ep$ for sufficiently small $\ep$. |
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232 |
|
342 | 233 |
We define a degree $1$ map $j_\ep: L_*^\ep \to L_*^\ep$ as follows. |
234 |
Let $x \in L_*^\ep$ be a blob diagram. |
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235 |
%\nn{maybe add figures illustrating $j_\ep$?} |
342 | 236 |
If $*$ is not contained in any twig blob, we define $j_\ep(x)$ by adding |
237 |
$N_\ep$ as a new twig blob, with label $y - s(y)$ where $y$ is the restriction |
|
238 |
of $x$ to $N_\ep$. |
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409 | 239 |
If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$, |
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240 |
%\nn{SM: I don't think we need to consider sums here} |
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241 |
%\nn{KW: It depends on whether we allow linear combinations of fields outside of twig blobs} |
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write $y_i$ for the restriction of $z_i$ to $N_\ep$, and let |
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$x_i$ be equal to $x$ on $S^1 \setmin B$, equal to $z_i$ on $B \setmin N_\ep$, |
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and have an additional blob $N_\ep$ with label $y_i - s(y_i)$. |
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Define $j_\ep(x) = \sum x_i$. |
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It is not hard to show that on $L_*^\ep$ |
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\[ |
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\bd j_\ep + j_\ep \bd = \id - i \circ s . |
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250 |
\] |
437 | 251 |
(To get the signs correct here, we add $N_\ep$ as the first blob.) |
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252 |
Since for $\ep$ small enough $L_*^\ep$ captures all of the |
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homology of $J_*$, |
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it follows that the mapping cone of $i \circ s$ is acyclic and therefore (using the fact that |
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255 |
these complexes are free) $i \circ s$ is homotopic to the identity. |
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256 |
\end{proof} |
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257 |
|
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\begin{proof}[Proof of Lemma \ref{lem:hochschild-exact}] |
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We now prove that $K_*$ is an exact functor. |
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|
232 | 261 |
As a warm-up, we prove |
262 |
that the functor on $C$-$C$ bimodules |
|
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263 |
\begin{equation*} |
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M \mapsto \ker(C \tensor M \tensor C \xrightarrow{c_1 \tensor m \tensor c_2 \mapsto c_1 m c_2} M) |
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\end{equation*} |
232 | 266 |
is exact. |
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267 |
Suppose we have a short exact sequence of $C$-$C$ bimodules $$\xymatrix{0 \ar[r] & K \ar@{^{(}->}[r]^f & E \ar@{->>}[r]^g & Q \ar[r] & 0}.$$ |
232 | 268 |
We'll write $\hat{f}$ and $\hat{g}$ for the image of $f$ and $g$ under the functor, so |
269 |
\[ |
|
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\hat{f}(\textstyle\sum_i a_i \tensor k_i \tensor b_i) = |
|
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\textstyle\sum_i a_i \tensor f(k_i) \tensor b_i , |
|
272 |
\] |
|
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and similarly for $\hat{g}$. |
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Most of what we need to check is easy. |
342 | 275 |
Suppose we have $\sum_i (a_i \tensor k_i \tensor b_i) \in \ker(C \tensor K \tensor C \to K)$, |
276 |
assuming without loss of generality that $\{a_i \tensor b_i\}_i$ is linearly independent in $C \tensor C$, |
|
277 |
and $\hat{f}(a \tensor k \tensor b) = 0 \in \ker(C \tensor E \tensor C \to E)$. |
|
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We must then have $f(k_i) = 0 \in E$ for each $i$, which implies $k_i=0$ itself. |
|
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If $\sum_i (a_i \tensor e_i \tensor b_i) \in \ker(C \tensor E \tensor C \to E)$ |
|
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is in the image of $\ker(C \tensor K \tensor C \to K)$ under $\hat{f}$, |
|
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again by assuming the set $\{a_i \tensor b_i\}_i$ is linearly independent we can deduce that each |
|
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$e_i$ is in the image of the original $f$, and so is in the kernel of the original $g$, |
|
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and so $\hat{g}(\sum_i a_i \tensor e_i \tensor b_i) = 0$. |
|
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If $\hat{g}(\sum_i a_i \tensor e_i \tensor b_i) = 0$, then each $g(e_i) = 0$, so $e_i = f(\widetilde{e_i})$ |
|
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for some $\widetilde{e_i} \in K$, and $\sum_i a_i \tensor e_i \tensor b_i = \hat{f}(\sum_i a_i \tensor \widetilde{e_i} \tensor b_i)$. |
|
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Finally, the interesting step is in checking that any $q = \sum_i a_i \tensor q_i \tensor b_i$ |
|
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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 $\hat{g}$. |
|
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For each $i$, we can find $\widetilde{q_i}$ so $g(\widetilde{q_i}) = q_i$. |
|
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However $\sum_i a_i \widetilde{q_i} b_i$ need not be zero. |
|
437 | 290 |
Consider then $$\widetilde{q} = \sum_i \left(a_i \tensor \widetilde{q_i} \tensor b_i\right) - 1 \tensor \left(\sum_i a_i \widetilde{q_i} b_i\right) \tensor 1.$$ Certainly |
342 | 291 |
$\widetilde{q} \in \ker(C \tensor E \tensor C \to E)$. |
292 |
Further, |
|
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293 |
\begin{align*} |
437 | 294 |
\hat{g}(\widetilde{q}) & = \sum_i \left(a_i \tensor g(\widetilde{q_i}) \tensor b_i\right) - 1 \tensor \left(\sum_i a_i g(\widetilde{q_i}\right) b_i) \tensor 1 \\ |
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& = q - 0 |
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296 |
\end{align*} |
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297 |
(here we used that $g$ is a map of $C$-$C$ bimodules, and that $\sum_i a_i q_i b_i = 0$). |
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298 |
|
69 | 299 |
Similar arguments show that the functors |
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300 |
\begin{equation} |
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301 |
\label{eq:ker-functor}% |
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M \mapsto \ker(C^{\tensor k} \tensor M \tensor C^{\tensor l} \to M) |
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\end{equation} |
342 | 304 |
are all exact too. |
305 |
Moreover, tensor products of such functors with each |
|
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other and with $C$ or $\ker(C^{\tensor k} \to C)$ (e.g., producing the functor $M \mapsto \ker(M \tensor C \to M) |
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\tensor C \tensor \ker(C \tensor C \to M)$) are all still exact. |
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308 |
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Finally, then we see that the functor $K_*$ is simply an (infinite) |
342 | 310 |
direct sum of copies of this sort of functor. |
311 |
The direct sum is indexed by |
|
312 |
configurations of nested blobs and of labels; for each such configuration, we have one of |
|
313 |
the above tensor product functors, |
|
314 |
with the labels of twig blobs corresponding to tensor factors as in \eqref{eq:ker-functor} |
|
315 |
or $\ker(C^{\tensor k} \to C)$ (depending on whether they contain a marked point $p_i$), and all other labelled points corresponding |
|
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316 |
to tensor factors of $C$ and $M$. |
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317 |
\end{proof} |
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318 |
\begin{proof}[Proof of Lemma \ref{lem:hochschild-coinvariants}] |
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319 |
We show that $H_0(K_*(M))$ is isomorphic to the coinvariants of $M$. |
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320 |
|
342 | 321 |
We define a map $\ev: K_0(M) \to M$. |
322 |
If $x \in K_0(M)$ has the label $m \in M$ at $*$, and labels $c_i \in C$ at the other |
|
323 |
labeled points of $S^1$, reading clockwise from $*$, |
|
324 |
we set $\ev(x) = m c_1 \cdots c_k$. |
|
325 |
We can think of this as $\ev : M \tensor C^{\tensor k} \to M$, for each direct summand of |
|
326 |
$K_0(M)$ indexed by a configuration of labeled points. |
|
232 | 327 |
|
328 |
There is a quotient map $\pi: M \to \coinv{M}$. |
|
329 |
We claim that the composition $\pi \compose \ev$ is well-defined on the quotient $H_0(K_*(M))$; |
|
330 |
i.e.\ that $\pi(\ev(\bd y)) = 0$ for all $y \in K_1(M)$. |
|
331 |
There are two cases, depending on whether the blob of $y$ contains the point *. |
|
332 |
If it doesn't, then |
|
342 | 333 |
suppose $y$ has label $m$ at $*$, labels $c_i$ at other labeled points outside the blob, |
334 |
and the field inside the blob is a sum, with the $j$-th term having |
|
335 |
labeled points $d_{j,i}$. |
|
336 |
Then $\sum_j d_{j,1} \tensor \cdots \tensor d_{j,k_j} \in \ker(\DirectSum_k C^{\tensor k} \to C)$, and so |
|
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$\ev(\bdy y) = 0$, because $$C^{\tensor \ell_1} \tensor \ker(\DirectSum_k C^{\tensor k} \to C) \tensor C^{\tensor \ell_2} \subset \ker(\DirectSum_k C^{\tensor k} \to C).$$ |
342 | 338 |
Similarly, if $*$ is contained in the blob, then the blob label is a sum, with the |
339 |
$j$-th term have labelled points $d_{j,i}$ to the left of $*$, $m_j$ at $*$, and $d_{j,i}'$ to the right of $*$, |
|
340 |
and there are labels $c_i$ at the labeled points outside the blob. |
|
341 |
We know that |
|
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$$\sum_j d_{j,1} \tensor \cdots \tensor d_{j,k_j} \tensor m_j \tensor d_{j,1}' \tensor \cdots \tensor d_{j,k'_j}' \in \ker(\DirectSum_{k,k'} C^{\tensor k} \tensor M \tensor C^{\tensor k'} \tensor \to M),$$ |
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and so |
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344 |
\begin{align*} |
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345 |
\ev(\bdy y) & = \sum_j m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k d_{j,1} \cdots d_{j,k_j} \\ |
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& = \sum_j d_{j,1} \cdots d_{j,k_j} m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k \\ |
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& = 0 |
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348 |
\end{align*} |
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349 |
where this time we use the fact that we're mapping to $\coinv{M}$, not just $M$. |
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350 |
|
342 | 351 |
The map $\pi \compose \ev: H_0(K_*(M)) \to \coinv{M}$ is clearly |
352 |
surjective ($\ev$ surjects onto $M$); we now show that it's injective. |
|
252 | 353 |
This is equivalent to showing that |
354 |
\[ |
|
355 |
\ev\inv(\ker(\pi)) \sub \bd K_1(M) . |
|
356 |
\] |
|
357 |
The above inclusion follows from |
|
358 |
\[ |
|
359 |
\ker(\ev) \sub \bd K_1(M) |
|
360 |
\] |
|
361 |
and |
|
362 |
\[ |
|
363 |
\ker(\pi) \sub \ev(\bd K_1(M)) . |
|
364 |
\] |
|
365 |
Let $x = \sum x_i$ be in the kernel of $\ev$, where each $x_i$ is a configuration of |
|
366 |
labeled points in $S^1$. |
|
367 |
Since the sum is finite, we can find an interval (blob) $B$ in $S^1$ |
|
368 |
such that for each $i$ the $C$-labeled points of $x_i$ all lie to the right of the |
|
369 |
base point *. |
|
370 |
Let $y_i$ be the restriction of $x_i$ to $B$ and $y = \sum y_i$. |
|
371 |
Let $r$ be the ``empty" field on $S^1 \setmin B$. |
|
372 |
It follows that $y \in U(B)$ and |
|
373 |
\[ |
|
374 |
\bd(B, y, r) = x . |
|
375 |
\] |
|
376 |
$\ker(\pi)$ is generated by elements of the form $cm - mc$. |
|
377 |
As shown in Figure \ref{fig:hochschild-1-chains}, $cm - mc$ lies in $\ev(\bd K_1(M))$. |
|
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378 |
\end{proof} |
252 | 379 |
|
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380 |
\begin{proof}[Proof of Lemma \ref{lem:hochschild-free}] |
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381 |
We show that $K_*(C\otimes C)$ is |
342 | 382 |
quasi-isomorphic to the 0-step complex $C$. |
383 |
We'll do this in steps, establishing quasi-isomorphisms and homotopy equivalences |
|
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384 |
$$K_*(C \tensor C) \quismto K'_* \htpyto K''_* \quismto C.$$ |
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385 |
|
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386 |
Let $K'_* \sub K_*(C\otimes C)$ be the subcomplex where the label of |
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387 |
the point $*$ is $1 \otimes 1 \in C\otimes C$. |
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388 |
We will show that the inclusion $i: K'_* \to K_*(C\otimes C)$ is a quasi-isomorphism. |
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389 |
|
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390 |
Fix a small $\ep > 0$. |
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391 |
Let $N_\ep$ be the ball of radius $\ep$ around $* \in S^1$. |
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392 |
Let $K_*^\ep \sub K_*(C\otimes C)$ be the subcomplex |
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393 |
generated by blob diagrams $b$ such that $N_\ep$ is either disjoint from |
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394 |
or contained in each blob of $b$, and the only labeled point inside $N_\ep$ is $*$. |
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%and the two boundary points of $N_\ep$ are not labeled points of $b$. |
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396 |
For a field $y$ on $N_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$ |
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397 |
labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$. |
342 | 398 |
(See Figure \ref{fig:sy}.) |
399 |
Note that $y - s_\ep(y) \in U(N_\ep)$. |
|
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400 |
Let $\sigma_\ep: K_*^\ep \to K_*^\ep$ be the chain map |
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401 |
given by replacing the restriction $y$ to $N_\ep$ of each field |
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402 |
appearing in an element of $K_*^\ep$ with $s_\ep(y)$. |
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403 |
Note that $\sigma_\ep(x) \in K'_*$. |
252 | 404 |
\begin{figure}[t] |
43
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405 |
\begin{align*} |
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406 |
y & = \mathfig{0.2}{hochschild/y} & |
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407 |
s_\ep(y) & = \mathfig{0.2}{hochschild/sy} |
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408 |
\end{align*} |
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409 |
\caption{Defining $s_\ep$.} |
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410 |
\label{fig:sy} |
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411 |
\end{figure} |
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412 |
|
232 | 413 |
Define a degree 1 map $j_\ep : K_*^\ep \to K_*^\ep$ as follows. |
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414 |
Let $x \in K_*^\ep$ be a blob diagram. |
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415 |
If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $N_\ep$ to |
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416 |
$x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $N_\ep$. |
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417 |
If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows. |
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418 |
Let $y_i$ be the restriction of $z_i$ to $N_\ep$. |
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419 |
Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin N_\ep$, |
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420 |
and have an additional blob $N_\ep$ with label $y_i - s_\ep(y_i)$. |
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421 |
Define $j_\ep(x) = \sum x_i$. |
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422 |
Note that if $x \in K'_* \cap K_*^\ep$ then $j_\ep(x) \in K'_*$ also. |
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423 |
|
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424 |
The key property of $j_\ep$ is |
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425 |
\eq{ |
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426 |
\bd j_\ep + j_\ep \bd = \id - \sigma_\ep. |
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427 |
} |
437 | 428 |
(Again, to get the correct signs, $N_\ep$ must be added as the first blob.) |
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429 |
If $j_\ep$ were defined on all of $K_*(C\otimes C)$, this would show that $\sigma_\ep$ |
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430 |
is a homotopy inverse to the inclusion $K'_* \to K_*(C\otimes C)$. |
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431 |
One strategy would be to try to stitch together various $j_\ep$ for progressively smaller |
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|
432 |
$\ep$ and show that $K'_*$ is homotopy equivalent to $K_*(C\otimes C)$. |
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433 |
Instead, we'll be less ambitious and just show that |
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434 |
$K'_*$ is quasi-isomorphic to $K_*(C\otimes C)$. |
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435 |
|
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436 |
If $x$ is a cycle in $K_*(C\otimes C)$, then for sufficiently small $\ep$ we have |
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437 |
$x \in K_*^\ep$. |
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438 |
(This is true for any chain in $K_*(C\otimes C)$, since chains are sums of |
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439 |
finitely many blob diagrams.) |
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440 |
Then $x$ is homologous to $\sigma_\ep(x)$, which is in $K'_*$, so the inclusion map |
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441 |
$K'_* \sub K_*(C\otimes C)$ is surjective on homology. |
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442 |
If $y \in K_*(C\otimes C)$ and $\bd y = x \in K_*(C\otimes C)$, then $y \in K_*^\ep$ for some $\ep$ |
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443 |
and |
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444 |
\eq{ |
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445 |
\bd y = \bd (\sigma_\ep(y) + j_\ep(x)) . |
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446 |
} |
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447 |
Since $\sigma_\ep(y) + j_\ep(x) \in K'_*$, it follows that the inclusion map is injective on homology. |
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448 |
This completes the proof that $K'_*$ is quasi-isomorphic to $K_*(C\otimes C)$. |
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449 |
|
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450 |
Let $K''_* \sub K'_*$ be the subcomplex of $K'_*$ where $*$ is not contained in any blob. |
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451 |
We will show that the inclusion $i: K''_* \to K'_*$ is a homotopy equivalence. |
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452 |
|
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453 |
First, a lemma: Let $G''_*$ and $G'_*$ be defined similarly to $K''_*$ and $K'_*$, except with |
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454 |
$S^1$ replaced by some neighborhood $N$ of $* \in S^1$. |
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455 |
($G''_*$ and $G'_*$ depend on $N$, but that is not reflected in the notation.) |
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456 |
Then $G''_*$ and $G'_*$ are both contractible |
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457 |
and the inclusion $G''_* \sub G'_*$ is a homotopy equivalence. |
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458 |
For $G'_*$ the proof is the same as in (\ref{bcontract}), except that the splitting |
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459 |
$G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$. |
257 | 460 |
For $G''_*$ we note that any cycle is supported away from $*$. |
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461 |
Thus any cycle lies in the image of the normal blob complex of a disjoint union |
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462 |
of two intervals, which is contractible by (\ref{bcontract}) and (\ref{disj-union-contract}). |
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463 |
Finally, it is easy to see that the inclusion |
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464 |
$G''_* \to G'_*$ induces an isomorphism on $H_0$. |
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465 |
|
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466 |
Next we construct a degree 1 map (homotopy) $h: K'_* \to K'_*$ such that |
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467 |
for all $x \in K'_*$ we have |
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468 |
\eq{ |
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469 |
x - \bd h(x) - h(\bd x) \in K''_* . |
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470 |
} |
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471 |
Since $K'_0 = K''_0$, we can take $h_0 = 0$. |
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472 |
Let $x \in K'_1$, with single blob $B \sub S^1$. |
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473 |
If $* \notin B$, then $x \in K''_1$ and we define $h_1(x) = 0$. |
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474 |
If $* \in B$, then we work in the image of $G'_*$ and $G''_*$ (with $B$ playing the role of $N$ above). |
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475 |
Choose $x'' \in G''_1$ such that $\bd x'' = \bd x$. |
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476 |
Since $G'_*$ is contractible, there exists $y \in G'_2$ such that $\bd y = x - x''$. |
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477 |
Define $h_1(x) = y$. |
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478 |
The general case is similar, except that we have to take lower order homotopies into account. |
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479 |
Let $x \in K'_k$. |
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480 |
If $*$ is not contained in any of the blobs of $x$, then define $h_k(x) = 0$. |
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481 |
Otherwise, let $B$ be the outermost blob of $x$ containing $*$. |
252 | 482 |
We can decompose $x = x' \bullet p$, |
483 |
where $x'$ is supported on $B$ and $p$ is supported away from $B$. |
|
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484 |
So $x' \in G'_l$ for some $l \le k$. |
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485 |
Choose $x'' \in G''_l$ such that $\bd x'' = \bd (x' - h_{l-1}\bd x')$. |
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486 |
Choose $y \in G'_{l+1}$ such that $\bd y = x' - x'' - h_{l-1}\bd x'$. |
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487 |
Define $h_k(x) = y \bullet p$. |
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488 |
This completes the proof that $i: K''_* \to K'_*$ is a homotopy equivalence. |
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489 |
%\nn{need to say above more clearly and settle on notation/terminology} |
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490 |
|
257 | 491 |
Finally, we show that $K''_*$ is contractible with $H_0\cong C$. |
492 |
This is similar to the proof of Proposition \ref{bcontract}, but a bit more |
|
493 |
complicated since there is no single blob which contains the support of all blob diagrams |
|
494 |
in $K''_*$. |
|
495 |
Let $x$ be a cycle of degree greater than zero in $K''_*$. |
|
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496 |
The union of the supports of the diagrams in $x$ does not contain $*$, so there exists a |
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497 |
ball $B \subset S^1$ containing the union of the supports and not containing $*$. |
257 | 498 |
Adding $B$ as an outermost blob to each summand of $x$ gives a chain $y$ with $\bd y = x$. |
499 |
Thus $H_i(K''_*) \cong 0$ for $i> 0$ and $K''_*$ is contractible. |
|
500 |
||
501 |
To see that $H_0(K''_*) \cong C$, consider the map $p: K''_0 \to C$ which sends a 0-blob |
|
502 |
diagram to the product of its labeled points. |
|
503 |
$p$ is clearly surjective. |
|
504 |
It's also easy to see that $p(\bd K''_1) = 0$. |
|
505 |
Finally, if $p(y) = 0$ then there exists a blob $B \sub S^1$ which contains |
|
506 |
all of the labeled points (other than *) of all of the summands of $y$. |
|
507 |
This allows us to construct $x\in K''_1$ such that $\bd x = y$. |
|
508 |
(The label of $B$ is the restriction of $y$ to $B$.) |
|
509 |
It follows that $H_0(K''_*) \cong C$. |
|
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|
510 |
\end{proof} |
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511 |
|
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|
512 |
\subsection{An explicit chain map in low degrees} |
74 | 513 |
|
514 |
For purposes of illustration, we describe an explicit chain map |
|
136 | 515 |
$\HC_*(M) \to K_*(M)$ |
74 | 516 |
between the Hochschild complex and the blob complex (with bimodule point) |
517 |
for degree $\le 2$. |
|
518 |
This map can be completed to a homotopy equivalence, though we will not prove that here. |
|
519 |
There are of course many such maps; what we describe here is one of the simpler possibilities. |
|
314 | 520 |
%Describing the extension to higher degrees is straightforward but tedious. |
521 |
%\nn{but probably we should include the general case in a future version of this paper} |
|
74 | 522 |
|
136 | 523 |
Recall that in low degrees $\HC_*(M)$ is |
74 | 524 |
\[ |
525 |
\cdots \stackrel{\bd}{\to} M \otimes C\otimes C \stackrel{\bd}{\to} |
|
526 |
M \otimes C \stackrel{\bd}{\to} M |
|
527 |
\] |
|
528 |
with |
|
529 |
\eqar{ |
|
530 |
\bd(m\otimes a) & = & ma - am \\ |
|
531 |
\bd(m\otimes a \otimes b) & = & ma\otimes b - m\otimes ab + bm \otimes a . |
|
532 |
} |
|
437 | 533 |
In degree 0, we send $m\in M$ to the 0-blob diagram $\mathfig{0.04}{hochschild/0-chains}$; the base point |
74 | 534 |
in $S^1$ is labeled by $m$ and there are no other labeled points. |
535 |
In degree 1, we send $m\ot a$ to the sum of two 1-blob diagrams |
|
77 | 536 |
as shown in Figure \ref{fig:hochschild-1-chains}. |
537 |
||
437 | 538 |
\begin{figure}[ht] |
77 | 539 |
\begin{equation*} |
540 |
\mathfig{0.4}{hochschild/1-chains} |
|
541 |
\end{equation*} |
|
542 |
\begin{align*} |
|
543 |
u_1 & = \mathfig{0.05}{hochschild/u_1-1} - \mathfig{0.05}{hochschild/u_1-2} & u_2 & = \mathfig{0.05}{hochschild/u_2-1} - \mathfig{0.05}{hochschild/u_2-2} |
|
544 |
\end{align*} |
|
545 |
\caption{The image of $m \tensor a$ in the blob complex.} |
|
546 |
\label{fig:hochschild-1-chains} |
|
547 |
\end{figure} |
|
548 |
||
437 | 549 |
\begin{figure}[ht] |
77 | 550 |
\begin{equation*} |
551 |
\mathfig{0.6}{hochschild/2-chains-0} |
|
552 |
\end{equation*} |
|
437 | 553 |
\caption{The 0-chains in the image of $m \tensor a \tensor b$.} |
554 |
\label{fig:hochschild-2-chains-0} |
|
555 |
\end{figure} |
|
556 |
\begin{figure}[ht] |
|
77 | 557 |
\begin{equation*} |
558 |
\mathfig{0.4}{hochschild/2-chains-1} \qquad \mathfig{0.4}{hochschild/2-chains-2} |
|
559 |
\end{equation*} |
|
437 | 560 |
\caption{The 1- and 2-chains in the image of $m \tensor a \tensor b$. |
561 |
Only the supports of the blobs are shown, but see Figure \ref{fig:hochschild-example-2-cell} for an example of a $2$-cell label.} |
|
562 |
\label{fig:hochschild-2-chains-12} |
|
77 | 563 |
\end{figure} |
74 | 564 |
|
437 | 565 |
\begin{figure}[ht] |
77 | 566 |
\begin{equation*} |
567 |
A = \mathfig{0.1}{hochschild/v_1} + \mathfig{0.1}{hochschild/v_2} + \mathfig{0.1}{hochschild/v_3} + \mathfig{0.1}{hochschild/v_4} |
|
568 |
\end{equation*} |
|
569 |
\begin{align*} |
|
570 |
v_1 & = \mathfig{0.05}{hochschild/v_1-1} - \mathfig{0.05}{hochschild/v_1-2} & v_2 & = \mathfig{0.05}{hochschild/v_2-1} - \mathfig{0.05}{hochschild/v_2-2} \\ |
|
571 |
v_3 & = \mathfig{0.05}{hochschild/v_3-1} - \mathfig{0.05}{hochschild/v_3-2} & v_4 & = \mathfig{0.05}{hochschild/v_4-1} - \mathfig{0.05}{hochschild/v_4-2} |
|
572 |
\end{align*} |
|
437 | 573 |
\caption{One of the 2-cells from Figure \ref{fig:hochschild-2-chains-12}.} |
77 | 574 |
\label{fig:hochschild-example-2-cell} |
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575 |
\end{figure} |
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576 |
|
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577 |
In degree 2, we send $m\ot a \ot b$ to the sum of 24 ($=6\cdot4$) 2-blob diagrams as shown in |
437 | 578 |
Figures \ref{fig:hochschild-2-chains-0} and \ref{fig:hochschild-2-chains-12}. |
579 |
In Figure \ref{fig:hochschild-2-chains-12} the 1- and 2-blob diagrams are indicated only by their support. |
|
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580 |
We leave it to the reader to determine the labels of the 1-blob diagrams. |
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581 |
Each 2-cell in the figure is labeled by a ball $V$ in $S^1$ which contains the support of all |
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582 |
1-blob diagrams in its boundary. |
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583 |
Such a 2-cell corresponds to a sum of the 2-blob diagrams obtained by adding $V$ |
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584 |
as an outer (non-twig) blob to each of the 1-blob diagrams in the boundary of the 2-cell. |
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585 |
Figure \ref{fig:hochschild-example-2-cell} shows this explicitly for the 2-cell |
437 | 586 |
labeled $A$ in Figure \ref{fig:hochschild-2-chains-12}. |
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587 |
Note that the (blob complex) boundary of this sum of 2-blob diagrams is |
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588 |
precisely the sum of the 1-blob diagrams corresponding to the boundary of the 2-cell. |
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589 |
(Compare with the proof of \ref{bcontract}.) |