blob: comparison text/hochschild.tex

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-:9a459c7f360e
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 generated by blob diagrams $b$ such that $N_\ep$ is either disjoint from
 or contained in each blob of $b$, and the only labeled point inside $N_\ep$ is $*$.
 %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 $y - s_\ep(y) \in U(N_\ep)$. We can think of
+(See Figure \ref{fig:sy}.) Note that $y - s_\ep(y) \in U(N_\ep)$.
-$\sigma_\ep$ as a chain map $K_*^\ep \to K_*^\ep$ given by replacing the restriction $y$ to $N_\ep$ of each field
+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 $s_\ep(y)$.
 Note that $\sigma_\ep(x) \in K'_*$.
 \begin{figure}[!ht]
 \begin{align*}
 y & = \mathfig{0.2}{hochschild/y} &
 If $x$ is a cycle in $K_*(C\otimes C)$, then for sufficiently small $\ep$ we have
 $x \in K_*^\ep$.
 (This is true for any chain in $K_*(C\otimes C)$, since chains are sums of
 finitely many blob diagrams.)
-Then $x$ is homologous to $s_\ep(x)$, which is in $K'_*$, so the inclusion map
+Then $x$ is homologous to $\sigma_\ep(x)$, which is in $K'_*$, so the inclusion map
 $K'_* \sub K_*(C\otimes C)$ is surjective on homology.
-If $y \in K_*(C\otimes C)$ and $\bd y = x \in K'_*$, then $y \in K_*^\ep$ for some $\ep$
+If $y \in K_*(C\otimes C)$ and $\bd y = x \in K_*(C\otimes C)$, then $y \in K_*^\ep$ for some $\ep$
 and
 \eq{
 \bd y = \bd (\sigma_\ep(y) + j_\ep(x)) .
 }
-Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology.
+Since $\sigma_\ep(y) + j_\ep(x) \in K'_*$, it follows that the inclusion map is injective on homology.
 This completes the proof that $K'_*$ is quasi-isomorphic to $K_*(C\otimes C)$.
 Let $K''_* \sub K'_*$ be the subcomplex of $K'_*$ where $*$ is not contained in any blob.
 We will show that the inclusion $i: K''_* \to K'_*$ is a homotopy equivalence.
-First, a lemma:  Let $G''_*$ and $G'_*$ be defined the same as $K''_*$ and $K'_*$, except with
+First, a lemma:  Let $G''_*$ and $G'_*$ be defined similarly to $K''_*$ and $K'_*$, except with
-$S^1$ replaced some (any) neighborhood of $* \in S^1$.
+$S^1$ replaced by some neighborhood $N$ of $* \in S^1$.
+($G''_*$ and $G'_*$ depend on $N$, but that is not reflected in the notation.)
 Then $G''_*$ and $G'_*$ are both contractible
 and the inclusion $G''_* \sub G'_*$ is a homotopy equivalence.
 For $G'_*$ the proof is the same as in (\ref{bcontract}), except that the splitting
 $G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$.
 For $G''_*$ we note that any cycle is supported \nn{need to establish terminology for this; maybe
 in ``basic properties" section above} away from $*$.
 Thus any cycle lies in the image of the normal blob complex of a disjoint union
-of two intervals, which is contractible by (\ref{bcontract}) and (\ref{disjunion}).
+of two intervals, which is contractible by (\ref{bcontract}) and (\ref{disj-union-contract}).
-Actually, we need the further (easy) result that the inclusion
+Finally, it is easy to see that the inclusion
 $G''_* \to G'_*$ induces an isomorphism on $H_0$.
 Next we construct a degree 1 map (homotopy) $h: K'_* \to K'_*$ such that
 for all $x \in K'_*$ we have
 \eq{
 \end{align*}
 \caption{The image of $m \tensor a$ in the blob complex.}
 \label{fig:hochschild-1-chains}
 \end{figure}
-In degree 2, we send $m\ot a \ot b$ to the sum of 24 (=6*4) 2-blob diagrams as shown in
+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.
 We leave it to the reader to determine the labels of the 1-blob diagrams.
 \begin{figure}[!ht]
 \begin{equation*}
 \mathfig{0.6}{hochschild/2-chains-0}
 \end{figure}
 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.
 Such a 2-cell corresponds to a sum of the 2-blob diagrams obtained by adding $V$
 as an outer (non-twig) blob to each of the 1-blob diagrams in the boundary of the 2-cell.
-Figure \ref{fig:hochschild-example-2-cell} shows this explicitly for one of the 2-cells.
+Figure \ref{fig:hochschild-example-2-cell} shows this explicitly for the 2-cell
+labeled $A$ in Figure \ref{fig:hochschild-2-chains}.
 Note that the (blob complex) boundary of this sum of 2-blob diagrams is
 precisely the sum of the 1-blob diagrams corresponding to the boundary of the 2-cell.
 (Compare with the proof of \ref{bcontract}.)
 \begin{figure}[!ht]

changeset 234	fa0ec034acc6
parent 232	9a459c7f360e
child 244	cf01e213044a