author | kevin@6e1638ff-ae45-0410-89bd-df963105f760 |
Sat, 05 Jul 2008 20:44:17 +0000 | |
changeset 33 | 0535a42fb804 |
parent 28 | f844cffa5c03 |
child 38 | 0a43a274744a |
permissions | -rw-r--r-- |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1 |
In this section we analyze the blob complex in dimension $n=1$ |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
2 |
and find that for $S^1$ the homology of the blob complex is the |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
3 |
Hochschild homology of the category (algebroid) that we started with. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
4 |
\nn{or maybe say here that the complexes are quasi-isomorphic? in general, |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
5 |
should perhaps put more emphasis on the complexes and less on the homology.} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
6 |
|
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
7 |
Notation: $HB_i(X) = H_i(\bc_*(X))$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
8 |
|
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
9 |
Let us first note that there is no loss of generality in assuming that our system of |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
10 |
fields comes from a category. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
11 |
(Or maybe (???) there {\it is} a loss of generality. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
12 |
Given any system of fields, $A(I; a, b) = \cC(I; a, b)/U(I; a, b)$ can be |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
13 |
thought of as the morphisms of a 1-category $C$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
14 |
More specifically, the objects of $C$ are $\cC(pt)$, the morphisms from $a$ to $b$ |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
15 |
are $A(I; a, b)$, and composition is given by gluing. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
16 |
If we instead take our fields to be $C$-pictures, the $\cC(pt)$ does not change |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
17 |
and neither does $A(I; a, b) = HB_0(I; a, b)$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
18 |
But what about $HB_i(I; a, b)$ for $i > 0$? |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
19 |
Might these higher blob homology groups be different? |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
20 |
Seems unlikely, but I don't feel like trying to prove it at the moment. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
21 |
In any case, we'll concentrate on the case of fields based on 1-category |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
22 |
pictures for the rest of this section.) |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
23 |
|
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
24 |
(Another question: $\bc_*(I)$ is an $A_\infty$-category. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
25 |
How general of an $A_\infty$-category is it? |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
26 |
Given an arbitrary $A_\infty$-category can one find fields and local relations so |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
27 |
that $\bc_*(I)$ is in some sense equivalent to the original $A_\infty$-category? |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
28 |
Probably not, unless we generalize to the case where $n$-morphisms are complexes.) |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
29 |
|
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
30 |
Continuing... |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
31 |
|
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
32 |
Let $C$ be a *-1-category. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
33 |
Then specializing the definitions from above to the case $n=1$ we have: |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
34 |
\begin{itemize} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
35 |
\item $\cC(pt) = \ob(C)$ . |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
36 |
\item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
37 |
Then an element of $\cC(R; c)$ is a collection of (transversely oriented) |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
38 |
points in the interior |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
39 |
of $R$, each labeled by a morphism of $C$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
40 |
The intervals between the points are labeled by objects of $C$, consistent with |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
41 |
the boundary condition $c$ and the domains and ranges of the point labels. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
42 |
\item There is an evaluation map $e: \cC(I; a, b) \to \mor(a, b)$ given by |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
43 |
composing the morphism labels of the points. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
44 |
Note that we also need the * of *-1-category here in order to make all the morphisms point |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
45 |
the same way. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
46 |
\item For $x \in \mor(a, b)$ let $\chi(x) \in \cC(I; a, b)$ be the field with a single |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
47 |
point (at some standard location) labeled by $x$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
48 |
Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
49 |
form $y - \chi(e(y))$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
50 |
Thus we can, if we choose, restrict the blob twig labels to things of this form. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
51 |
\end{itemize} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
52 |
|
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
53 |
We want to show that $HB_*(S^1)$ is naturally isomorphic to the |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
54 |
Hochschild homology of $C$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
55 |
\nn{Or better that the complexes are homotopic |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
56 |
or quasi-isomorphic.} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
57 |
In order to prove this we will need to extend the blob complex to allow points to also |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
58 |
be labeled by elements of $C$-$C$-bimodules. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
59 |
%Given an interval (1-ball) so labeled, there is an evaluation map to some tensor product |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
60 |
%(over $C$) of $C$-$C$-bimodules. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
61 |
%Define the local relations $U(I; a, b)$ to be the direct sum of the kernels of these maps. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
62 |
%Now we can define the blob complex for $S^1$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
63 |
%This complex is the sum of complexes with a fixed cyclic tuple of bimodules present. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
64 |
%If $M$ is a $C$-$C$-bimodule, let $G_*(M)$ denote the summand of $\bc_*(S^1)$ corresponding |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
65 |
%to the cyclic 1-tuple $(M)$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
66 |
%In other words, $G_*(M)$ is a blob-like complex where exactly one point is labeled |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
67 |
%by an element of $M$ and the remaining points are labeled by morphisms of $C$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
68 |
%It's clear that $G_*(C)$ is isomorphic to the original bimodule-less |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
69 |
%blob complex for $S^1$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
70 |
%\nn{Is it really so clear? Should say more.} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
71 |
|
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
72 |
%\nn{alternative to the above paragraph:} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
73 |
Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$. |
19 | 74 |
We define a blob-like complex $K_*(S^1, (p_i), (M_i))$. |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
75 |
The fields have elements of $M_i$ labeling $p_i$ and elements of $C$ labeling |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
76 |
other points. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
77 |
The blob twig labels lie in kernels of evaluation maps. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
78 |
(The range of these evaluation maps is a tensor product (over $C$) of $M_i$'s.) |
19 | 79 |
Let $K_*(M) = K_*(S^1, (*), (M))$, where $* \in S^1$ is some standard base point. |
80 |
In other words, fields for $K_*(M)$ have an element of $M$ at the fixed point $*$ |
|
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
81 |
and elements of $C$ at variable other points. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
82 |
|
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
83 |
\todo{Some orphaned questions:} |
19 | 84 |
\nn{Or maybe we should claim that $M \to K_*(M)$ is the/a derived coend. |
85 |
Or maybe that $K_*(M)$ is quasi-isomorphic (or perhaps homotopic) to the Hochschild |
|
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
86 |
complex of $M$.} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
87 |
|
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
88 |
\nn{What else needs to be said to establish quasi-isomorphism to Hochschild complex? |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
89 |
Do we need a map from hoch to blob? |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
90 |
Does the above exactness and contractibility guarantee such a map without writing it |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
91 |
down explicitly? |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
92 |
Probably it's worth writing down an explicit map even if we don't need to.} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
93 |
|
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
94 |
|
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
95 |
We claim that |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
96 |
\begin{thm} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
97 |
The blob complex $\bc_*(S^1; C)$ on the circle is quasi-isomorphic to the |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
98 |
usual Hochschild complex for $C$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
99 |
\end{thm} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
100 |
|
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
101 |
This follows from two results. First, we see that |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
102 |
\begin{lem} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
103 |
\label{lem:module-blob}% |
19 | 104 |
The complex $K_*(C)$ (here $C$ is being thought of as a |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
105 |
$C$-$C$-bimodule, not a category) is quasi-isomorphic to the blob complex |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
106 |
$\bc_*(S^1; C)$. (Proof later.) |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
107 |
\end{lem} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
108 |
|
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
109 |
Next, we show that for any $C$-$C$-bimodule $M$, |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
110 |
\begin{prop} |
19 | 111 |
The complex $K_*(M)$ is quasi-isomorphic to $HC_*(M)$, the usual |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
112 |
Hochschild complex of $M$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
113 |
\end{prop} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
114 |
\begin{proof} |
28 | 115 |
Recall that the usual Hochschild complex of $M$ is uniquely determined, |
116 |
up to quasi-isomorphism, by the following properties: |
|
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
117 |
\begin{enumerate} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
118 |
\item \label{item:hochschild-additive}% |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
119 |
$HC_*(M_1 \oplus M_2) \cong HC_*(M_1) \oplus HC_*(M_2)$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
120 |
\item \label{item:hochschild-exact}% |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
121 |
An exact sequence $0 \to M_1 \into M_2 \onto M_3 \to 0$ gives rise to an |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
122 |
exact sequence $0 \to HC_*(M_1) \into HC_*(M_2) \onto HC_*(M_3) \to 0$. |
28 | 123 |
\item \label{item:hochschild-coinvariants}% |
124 |
$HH_0(M)$ is isomorphic to the coinvariants of $M$, $\coinv(M) = |
|
125 |
M/\langle cm-mc \rangle$. |
|
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
126 |
\item \label{item:hochschild-free}% |
28 | 127 |
$HC_*(C\otimes C)$ (the free $C$-$C$-bimodule with one generator) is contractible; that is, |
128 |
quasi-isomorphic to its $0$-th homology (which in turn, by \ref{item:hochschild-coinvariants}, is just $C$) via the quotient map $HC_0 \onto HH_0$. |
|
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
129 |
\end{enumerate} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
130 |
(Together, these just say that Hochschild homology is `the derived functor of coinvariants'.) |
19 | 131 |
We'll first recall why these properties are characteristic. |
132 |
||
133 |
Take some $C$-$C$ bimodule $M$, and choose a free resolution |
|
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
134 |
\begin{equation*} |
19 | 135 |
\cdots \to F_2 \xrightarrow{f_2} F_1 \xrightarrow{f_1} F_0. |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
136 |
\end{equation*} |
28 | 137 |
We will show that for any functor $\cP$ satisfying properties |
138 |
\ref{item:hochschild-additive}, \ref{item:hochschild-exact}, |
|
139 |
\ref{item:hochschild-coinvariants} and \ref{item:hochschild-free}, there |
|
140 |
is a quasi-isomorphism |
|
141 |
$$\cP_*(M) \iso \coinv(F_*).$$ |
|
142 |
% |
|
143 |
Observe that there's a quotient map $\pi: F_0 \onto M$, and by |
|
144 |
construction the cone of the chain map $\pi: F_* \to M$ is acyclic. Now |
|
145 |
construct the total complex $\cP_i(F_j)$, with $i,j \geq 0$, graded by |
|
146 |
$i+j$. We have two chain maps |
|
19 | 147 |
\begin{align*} |
28 | 148 |
\cP_i(F_*) & \xrightarrow{\cP_i(\pi)} \cP_i(M) \\ |
19 | 149 |
\intertext{and} |
28 | 150 |
\cP_*(F_j) & \xrightarrow{\cP_0(F_j) \onto H_0(\cP_*(F_j))} \coinv(F_j). |
19 | 151 |
\end{align*} |
152 |
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. |
|
153 |
In the second case, this is because the `columns' indexed by $j$ are acyclic, since $F_j$ is free. |
|
154 |
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 |
|
28 | 155 |
$$\cP_*(M) \quismto \coinv(F_*).$$ |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
156 |
|
19 | 157 |
%If $M$ is free, that is, a direct sum of copies of |
158 |
%$C \tensor C$, then properties \ref{item:hochschild-additive} and |
|
159 |
%\ref{item:hochschild-free} determine $HC_*(M)$. Otherwise, choose some |
|
160 |
%free cover $F \onto M$, and define $K$ to be this map's kernel. Thus we |
|
161 |
%have a short exact sequence $0 \to K \into F \onto M \to 0$, and hence a |
|
162 |
%short exact sequence of complexes $0 \to HC_*(K) \into HC_*(F) \onto HC_*(M) |
|
163 |
%\to 0$. Such a sequence gives a long exact sequence on homology |
|
164 |
%\begin{equation*} |
|
165 |
%%\begin{split} |
|
166 |
%\cdots \to HH_{i+1}(F) \to HH_{i+1}(M) \to HH_i(K) \to HH_i(F) \to \cdots % \\ |
|
167 |
%%\cdots \to HH_1(F) \to HH_1(M) \to HH_0(K) \to HH_0(F) \to HH_0(M). |
|
168 |
%%\end{split} |
|
169 |
%\end{equation*} |
|
170 |
%For any $i \geq 1$, $HH_{i+1}(F) = HH_i(F) = 0$, by properties |
|
171 |
%\ref{item:hochschild-additive} and \ref{item:hochschild-free}, and so |
|
172 |
%$HH_{i+1}(M) \iso HH_i(F)$. For $i=0$, \todo{}. |
|
173 |
% |
|
174 |
%This tells us how to |
|
175 |
%compute every homology group of $HC_*(M)$; we already know $HH_0(M)$ |
|
176 |
%(it's just coinvariants, by property \ref{item:hochschild-coinvariants}), |
|
177 |
%and higher homology groups are determined by lower ones in $HC_*(K)$, and |
|
178 |
%hence recursively as coinvariants of some other bimodule. |
|
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
179 |
|
19 | 180 |
The proposition then follows from the following lemmas, establishing that $K_*$ has precisely these required properties. |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
181 |
\begin{lem} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
182 |
\label{lem:hochschild-additive}% |
19 | 183 |
Directly from the definition, $K_*(M_1 \oplus M_2) \cong K_*(M_1) \oplus K_*(M_2)$. |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
184 |
\end{lem} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
185 |
\begin{lem} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
186 |
\label{lem:hochschild-exact}% |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
187 |
An exact sequence $0 \to M_1 \into M_2 \onto M_3 \to 0$ gives rise to an |
19 | 188 |
exact sequence $0 \to K_*(M_1) \into K_*(M_2) \onto K_*(M_3) \to 0$. |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
189 |
\end{lem} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
190 |
\begin{lem} |
28 | 191 |
\label{lem:hochschild-coinvariants}% |
192 |
$H_0(K_*(M))$ is isomorphic to the coinvariants of $M$. |
|
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
193 |
\end{lem} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
194 |
\begin{lem} |
28 | 195 |
\label{lem:hochschild-free}% |
196 |
$K_*(C\otimes C)$ is quasi-isomorphic to $H_0(K_*(C \otimes C)) \iso C$. |
|
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
197 |
\end{lem} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
198 |
|
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
199 |
The remainder of this section is devoted to proving Lemmas |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
200 |
\ref{lem:module-blob}, |
28 | 201 |
\ref{lem:hochschild-exact}, \ref{lem:hochschild-coinvariants} and |
202 |
\ref{lem:hochschild-free}. |
|
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
203 |
\end{proof} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
204 |
|
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
205 |
\begin{proof}[Proof of Lemma \ref{lem:module-blob}] |
19 | 206 |
We show that $K_*(C)$ is quasi-isomorphic to $\bc_*(S^1)$. |
207 |
$K_*(C)$ differs from $\bc_*(S^1)$ only in that the base point * |
|
208 |
is always a labeled point in $K_*(C)$, while in $\bc_*(S^1)$ it may or may not be. |
|
209 |
In other words, there is an inclusion map $i: K_*(C) \to \bc_*(S^1)$. |
|
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
210 |
|
28 | 211 |
We define a left inverse $s: \bc_*(S^1) \to K_*(C)$ to the inclusion as follows. |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
212 |
If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
213 |
* is a labeled point in $y$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
214 |
Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
215 |
Let $x \in \bc_*(S^1)$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
216 |
Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
217 |
$x$ with $y$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
218 |
It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
219 |
|
19 | 220 |
Let $L_*^\ep \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
221 |
in a neighborhood $B_\ep$ of *, except perhaps *. |
19 | 222 |
Note that for any chain $x \in \bc_*(S^1)$, $x \in L_*^\ep$ for sufficiently small $\ep$. |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
223 |
\nn{rest of argument goes similarly to above} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
224 |
\end{proof} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
225 |
\begin{proof}[Proof of Lemma \ref{lem:hochschild-exact}] |
28 | 226 |
\todo{p. 1478 of scott's notes} |
227 |
Essentially, this comes down to the unsurprising fact that the functor on $C$-$C$ bimodules |
|
228 |
\begin{equation*} |
|
229 |
M \mapsto \ker(C \tensor M \tensor C \xrightarrow{c_1 \tensor m \tensor c_2 \mapsto c_1 m c_2} M) |
|
230 |
\end{equation*} |
|
231 |
is exact. For completeness we'll explain this below. |
|
232 |
||
233 |
Suppose we have a short exact sequence of $C$-$C$ bimodules $$\xymatrix{0 \ar[r] & K \ar@{^{(}->}[r]^f & E \ar@{->>}[r]^g & Q \ar[r] & 0}.$$ |
|
234 |
We'll write $\hat{f}$ and $\hat{g}$ for the image of $f$ and $g$ under the functor. |
|
235 |
Most of what we need to check is easy. |
|
236 |
If $(a \tensor k \tensor b) \in \ker(C \tensor K \tensor C \to K)$, to have $\hat{f}(a \tensor k \tensor b) = 0 \in \ker(C \tensor E \tensor C \to E)$, we must have $f(k) = 0 \in E$, so |
|
237 |
be $k=0$ itself. If $(a \tensor e \tensor b) \in \ker(C \tensor E \tensor C \to E)$ is in the image of $\ker(C \tensor K \tensor C \to K)$ under $\hat{f}$, clearly |
|
238 |
$e$ is in the image of the original $f$, so is in the kernel of the original $g$, and so $\hat{g}(a \tensor e \tensor b) = 0$. |
|
239 |
If $\hat{g}(a \tensor e \tensor b) = 0$, then $g(e) = 0$, so $e = f(\widetilde{e})$ for some $\widetilde{e} \in K$, and $a \tensor e \tensor b = \hat{f}(a \tensor \widetilde{e} \tensor b)$. |
|
240 |
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 $\hat{g}$. |
|
241 |
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. |
|
242 |
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 |
|
243 |
$\widetilde{q} \in \ker(C \tensor E \tensor C \to E)$. Further, |
|
244 |
\begin{align*} |
|
245 |
\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 \\ |
|
246 |
& = q - 0 |
|
247 |
\end{align*} |
|
248 |
(here we used that $g$ is a map of $C$-$C$ bimodules, and that $\sum_i a_i q_i b_i = 0$). |
|
249 |
||
250 |
Identical arguments show that the functors |
|
251 |
\begin{equation*} |
|
252 |
M \mapsto \ker(C^{\tensor k} \tensor M \tensor C^{\tensor l} \to M) |
|
253 |
\end{equation*} |
|
254 |
are all exact too. |
|
255 |
||
256 |
Finally, then \todo{explain why this is all we need.} |
|
257 |
\end{proof} |
|
258 |
\begin{proof}[Proof of Lemma \ref{lem:hochschild-coinvariants}] |
|
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
259 |
\todo{} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
260 |
\end{proof} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
261 |
\begin{proof}[Proof of Lemma \ref{lem:hochschild-free}] |
19 | 262 |
We show that $K_*(C\otimes C)$ is |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
263 |
quasi-isomorphic to the 0-step complex $C$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
264 |
|
19 | 265 |
Let $K'_* \sub K_*(C\otimes C)$ be the subcomplex where the label of |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
266 |
the point $*$ is $1 \otimes 1 \in C\otimes C$. |
19 | 267 |
We will show that the inclusion $i: K'_* \to K_*(C\otimes C)$ is a quasi-isomorphism. |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
268 |
|
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
269 |
Fix a small $\ep > 0$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
270 |
Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$. |
19 | 271 |
Let $K_*^\ep \sub K_*(C\otimes C)$ be the subcomplex |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
272 |
generated by blob diagrams $b$ such that $B_\ep$ is either disjoint from |
28 | 273 |
or contained in each blob of $b$, and the only labeled point inside $B_\ep$ is $*$. |
274 |
%and the two boundary points of $B_\ep$ are not labeled points of $b$. |
|
275 |
For a field $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$ |
|
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
276 |
labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$. |
28 | 277 |
(See Figure \ref{fig:sy}.) Note that $y - s_\ep(y) \in U(B_\ep)$. We can think of |
278 |
$\sigma_\ep$ as a chain map $K_*^\ep \to K_*^\ep$ given by replacing the restriction $y$ to $B_\ep$ of each field |
|
279 |
appearing in an element of $K_*^\ep$ with $s_\ep(y)$. |
|
280 |
Note that $\sigma_\ep(x) \in K'_*$. |
|
281 |
\begin{figure}[!ht] |
|
282 |
\begin{align*} |
|
283 |
y & = \mathfig{0.2}{hochschild/y} & |
|
284 |
s_\ep(y) & = \mathfig{0.2}{hochschild/sy} |
|
285 |
\end{align*} |
|
286 |
\caption{Defining $s_\ep$.} |
|
287 |
\label{fig:sy} |
|
288 |
\end{figure} |
|
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
289 |
|
19 | 290 |
Define a degree 1 chain map $j_\ep : K_*^\ep \to K_*^\ep$ as follows. |
291 |
Let $x \in K_*^\ep$ be a blob diagram. |
|
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
292 |
If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $B_\ep$ to |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
293 |
$x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
294 |
If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
295 |
Let $y_i$ be the restriction of $z_i$ to $B_\ep$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
296 |
Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$, |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
297 |
and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
298 |
Define $j_\ep(x) = \sum x_i$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
299 |
\nn{need to check signs coming from blob complex differential} |
19 | 300 |
Note that if $x \in K'_* \cap K_*^\ep$ then $j_\ep(x) \in K'_*$ also. |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
301 |
|
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
302 |
The key property of $j_\ep$ is |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
303 |
\eq{ |
28 | 304 |
\bd j_\ep + j_\ep \bd = \id - \sigma_\ep. |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
305 |
} |
28 | 306 |
If $j_\ep$ were defined on all of $K_*(C\otimes C)$, this would show that $\sigma_\ep$ |
19 | 307 |
is a homotopy inverse to the inclusion $K'_* \to K_*(C\otimes C)$. |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
308 |
One strategy would be to try to stitch together various $j_\ep$ for progressively smaller |
19 | 309 |
$\ep$ and show that $K'_*$ is homotopy equivalent to $K_*(C\otimes C)$. |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
310 |
Instead, we'll be less ambitious and just show that |
19 | 311 |
$K'_*$ is quasi-isomorphic to $K_*(C\otimes C)$. |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
312 |
|
19 | 313 |
If $x$ is a cycle in $K_*(C\otimes C)$, then for sufficiently small $\ep$ we have |
314 |
$x \in K_*^\ep$. |
|
315 |
(This is true for any chain in $K_*(C\otimes C)$, since chains are sums of |
|
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
316 |
finitely many blob diagrams.) |
19 | 317 |
Then $x$ is homologous to $s_\ep(x)$, which is in $K'_*$, so the inclusion map |
318 |
$K'_* \sub K_*(C\otimes C)$ is surjective on homology. |
|
319 |
If $y \in K_*(C\otimes C)$ and $\bd y = x \in K'_*$, then $y \in K_*^\ep$ for some $\ep$ |
|
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
320 |
and |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
321 |
\eq{ |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
322 |
\bd y = \bd (\sigma_\ep(y) + j_\ep(x)) . |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
323 |
} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
324 |
Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology. |
19 | 325 |
This completes the proof that $K'_*$ is quasi-isomorphic to $K_*(C\otimes C)$. |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
326 |
|
19 | 327 |
Let $K''_* \sub K'_*$ be the subcomplex of $K'_*$ where $*$ is not contained in any blob. |
328 |
We will show that the inclusion $i: K''_* \to K'_*$ is a homotopy equivalence. |
|
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
329 |
|
19 | 330 |
First, a lemma: Let $G''_*$ and $G'_*$ be defined the same as $K''_*$ and $K'_*$, except with |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
331 |
$S^1$ replaced some (any) neighborhood of $* \in S^1$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
332 |
Then $G''_*$ and $G'_*$ are both contractible |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
333 |
and the inclusion $G''_* \sub G'_*$ is a homotopy equivalence. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
334 |
For $G'_*$ the proof is the same as in (\ref{bcontract}), except that the splitting |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
335 |
$G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
336 |
For $G''_*$ we note that any cycle is supported \nn{need to establish terminology for this; maybe |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
337 |
in ``basic properties" section above} away from $*$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
338 |
Thus any cycle lies in the image of the normal blob complex of a disjoint union |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
339 |
of two intervals, which is contractible by (\ref{bcontract}) and (\ref{disjunion}). |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
340 |
Actually, we need the further (easy) result that the inclusion |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
341 |
$G''_* \to G'_*$ induces an isomorphism on $H_0$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
342 |
|
19 | 343 |
Next we construct a degree 1 map (homotopy) $h: K'_* \to K'_*$ such that |
344 |
for all $x \in K'_*$ we have |
|
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
345 |
\eq{ |
19 | 346 |
x - \bd h(x) - h(\bd x) \in K''_* . |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
347 |
} |
19 | 348 |
Since $K'_0 = K''_0$, we can take $h_0 = 0$. |
349 |
Let $x \in K'_1$, with single blob $B \sub S^1$. |
|
350 |
If $* \notin B$, then $x \in K''_1$ and we define $h_1(x) = 0$. |
|
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
351 |
If $* \in B$, then we work in the image of $G'_*$ and $G''_*$ (with respect to $B$). |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
352 |
Choose $x'' \in G''_1$ such that $\bd x'' = \bd x$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
353 |
Since $G'_*$ is contractible, there exists $y \in G'_2$ such that $\bd y = x - x''$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
354 |
Define $h_1(x) = y$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
355 |
The general case is similar, except that we have to take lower order homotopies into account. |
19 | 356 |
Let $x \in K'_k$. |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
357 |
If $*$ is not contained in any of the blobs of $x$, then define $h_k(x) = 0$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
358 |
Otherwise, let $B$ be the outermost blob of $x$ containing $*$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
359 |
By xxxx above, $x = x' \bullet p$, where $x'$ is supported on $B$ and $p$ is supported away from $B$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
360 |
So $x' \in G'_l$ for some $l \le k$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
361 |
Choose $x'' \in G''_l$ such that $\bd x'' = \bd (x' - h_{l-1}\bd x')$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
362 |
Choose $y \in G'_{l+1}$ such that $\bd y = x' - x'' - h_{l-1}\bd x'$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
363 |
Define $h_k(x) = y \bullet p$. |
19 | 364 |
This completes the proof that $i: K''_* \to K'_*$ is a homotopy equivalence. |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
365 |
\nn{need to say above more clearly and settle on notation/terminology} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
366 |
|
19 | 367 |
Finally, we show that $K''_*$ is contractible. |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
368 |
\nn{need to also show that $H_0$ is the right thing; easy, but I won't do it now} |
19 | 369 |
Let $x$ be a cycle in $K''_*$. |
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
370 |
The union of the supports of the diagrams in $x$ does not contain $*$, so there exists a |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
371 |
ball $B \subset S^1$ containing the union of the supports and not containing $*$. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
372 |
Adding $B$ as a blob to $x$ gives a contraction. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
373 |
\nn{need to say something else in degree zero} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
374 |
\end{proof} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
375 |
|
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
376 |
We can also describe explicitly a map from the standard Hochschild |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
377 |
complex to the blob complex on the circle. \nn{What properties does this |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
378 |
map have?} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
379 |
|
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
380 |
\begin{figure}% |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
381 |
$$\mathfig{0.6}{barycentric/barycentric}$$ |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
382 |
\caption{The Hochschild chain $a \tensor b \tensor c$ is sent to |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
383 |
the sum of six blob $2$-chains, corresponding to a barycentric subdivision of a $2$-simplex.} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
384 |
\label{fig:Hochschild-example}% |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
385 |
\end{figure} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
386 |
|
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
387 |
As an example, Figure \ref{fig:Hochschild-example} shows the image of the Hochschild chain $a \tensor b \tensor c$. Only the $0$-cells are shown explicitly. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
388 |
The edges marked $x, y$ and $z$ carry the $1$-chains |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
389 |
\begin{align*} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
390 |
x & = \mathfig{0.1}{barycentric/ux} & u_x = \mathfig{0.1}{barycentric/ux_ca} - \mathfig{0.1}{barycentric/ux_c-a} \\ |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
391 |
y & = \mathfig{0.1}{barycentric/uy} & u_y = \mathfig{0.1}{barycentric/uy_cab} - \mathfig{0.1}{barycentric/uy_ca-b} \\ |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
392 |
z & = \mathfig{0.1}{barycentric/uz} & u_z = \mathfig{0.1}{barycentric/uz_c-a-b} - \mathfig{0.1}{barycentric/uz_cab} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
393 |
\end{align*} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
394 |
and the $2$-chain labelled $A$ is |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
395 |
\begin{equation*} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
396 |
A = \mathfig{0.1}{barycentric/Ax}+\mathfig{0.1}{barycentric/Ay}. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
397 |
\end{equation*} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
398 |
Note that we then have |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
399 |
\begin{equation*} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
400 |
\bdy A = x+y+z. |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
401 |
\end{equation*} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
402 |
|
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
403 |
In general, the Hochschild chain $\Tensor_{i=1}^n a_i$ is sent to the sum of $n!$ blob $(n-1)$-chains, indexed by permutations, |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
404 |
$$\phi\left(\Tensor_{i=1}^n a_i\right) = \sum_{\pi} \phi^\pi(a_1, \ldots, a_n)$$ |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
405 |
with ... (hmmm, problems making this precise; you need to decide where to put the labels, but then it's hard to make an honest chain map!) |