575 |
575 |
576 \bigskip |
576 \bigskip |
577 |
577 |
578 \nn{what else?} |
578 \nn{what else?} |
579 |
579 |
580 |
580 \section{Hochschild homology when $n=1$} |
581 |
581 \label{sec:hochschild} |
582 |
582 \input{text/hochschild} |
583 \section{$n=1$ and Hochschild homology} |
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584 |
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585 In this section we analyze the blob complex in dimension $n=1$ |
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586 and find that for $S^1$ the homology of the blob complex is the |
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587 Hochschild homology of the category (algebroid) that we started with. |
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588 \nn{or maybe say here that the complexes are quasi-isomorphic? in general, |
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589 should perhaps put more emphasis on the complexes and less on the homology.} |
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590 |
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591 Notation: $HB_i(X) = H_i(\bc_*(X))$. |
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592 |
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593 Let us first note that there is no loss of generality in assuming that our system of |
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594 fields comes from a category. |
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595 (Or maybe (???) there {\it is} a loss of generality. |
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596 Given any system of fields, $A(I; a, b) = \cC(I; a, b)/U(I; a, b)$ can be |
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597 thought of as the morphisms of a 1-category $C$. |
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598 More specifically, the objects of $C$ are $\cC(pt)$, the morphisms from $a$ to $b$ |
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599 are $A(I; a, b)$, and composition is given by gluing. |
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600 If we instead take our fields to be $C$-pictures, the $\cC(pt)$ does not change |
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601 and neither does $A(I; a, b) = HB_0(I; a, b)$. |
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602 But what about $HB_i(I; a, b)$ for $i > 0$? |
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603 Might these higher blob homology groups be different? |
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604 Seems unlikely, but I don't feel like trying to prove it at the moment. |
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605 In any case, we'll concentrate on the case of fields based on 1-category |
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606 pictures for the rest of this section.) |
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607 |
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608 (Another question: $\bc_*(I)$ is an $A_\infty$-category. |
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609 How general of an $A_\infty$-category is it? |
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610 Given an arbitrary $A_\infty$-category can one find fields and local relations so |
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611 that $\bc_*(I)$ is in some sense equivalent to the original $A_\infty$-category? |
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612 Probably not, unless we generalize to the case where $n$-morphisms are complexes.) |
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613 |
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614 Continuing... |
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615 |
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616 Let $C$ be a *-1-category. |
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617 Then specializing the definitions from above to the case $n=1$ we have: |
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618 \begin{itemize} |
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619 \item $\cC(pt) = \ob(C)$ . |
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620 \item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$. |
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621 Then an element of $\cC(R; c)$ is a collection of (transversely oriented) |
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622 points in the interior |
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623 of $R$, each labeled by a morphism of $C$. |
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624 The intervals between the points are labeled by objects of $C$, consistent with |
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625 the boundary condition $c$ and the domains and ranges of the point labels. |
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626 \item There is an evaluation map $e: \cC(I; a, b) \to \mor(a, b)$ given by |
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627 composing the morphism labels of the points. |
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628 Note that we also need the * of *-1-category here in order to make all the morphisms point |
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629 the same way. |
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630 \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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631 point (at some standard location) labeled by $x$. |
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632 Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the |
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633 form $y - \chi(e(y))$. |
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634 Thus we can, if we choose, restrict the blob twig labels to things of this form. |
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635 \end{itemize} |
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636 |
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637 We want to show that $HB_*(S^1)$ is naturally isomorphic to the |
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638 Hochschild homology of $C$. |
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639 \nn{Or better that the complexes are homotopic |
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640 or quasi-isomorphic.} |
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641 In order to prove this we will need to extend the blob complex to allow points to also |
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642 be labeled by elements of $C$-$C$-bimodules. |
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643 %Given an interval (1-ball) so labeled, there is an evaluation map to some tensor product |
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644 %(over $C$) of $C$-$C$-bimodules. |
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645 %Define the local relations $U(I; a, b)$ to be the direct sum of the kernels of these maps. |
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646 %Now we can define the blob complex for $S^1$. |
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647 %This complex is the sum of complexes with a fixed cyclic tuple of bimodules present. |
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648 %If $M$ is a $C$-$C$-bimodule, let $G_*(M)$ denote the summand of $\bc_*(S^1)$ corresponding |
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649 %to the cyclic 1-tuple $(M)$. |
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650 %In other words, $G_*(M)$ is a blob-like complex where exactly one point is labeled |
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651 %by an element of $M$ and the remaining points are labeled by morphisms of $C$. |
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652 %It's clear that $G_*(C)$ is isomorphic to the original bimodule-less |
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653 %blob complex for $S^1$. |
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654 %\nn{Is it really so clear? Should say more.} |
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655 |
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656 %\nn{alternative to the above paragraph:} |
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657 Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$. |
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658 We define a blob-like complex $F_*(S^1, (p_i), (M_i))$. |
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659 The fields have elements of $M_i$ labeling $p_i$ and elements of $C$ labeling |
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660 other points. |
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661 The blob twig labels lie in kernels of evaluation maps. |
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662 (The range of these evaluation maps is a tensor product (over $C$) of $M_i$'s.) |
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663 Let $F_*(M) = F_*(S^1, (*), (M))$, where $* \in S^1$ is some standard base point. |
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664 In other words, fields for $F_*(M)$ have an element of $M$ at the fixed point $*$ |
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665 and elements of $C$ at variable other points. |
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666 |
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667 We claim that the homology of $F_*(M)$ is isomorphic to the Hochschild |
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668 homology of $M$. |
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669 \nn{Or maybe we should claim that $M \to F_*(M)$ is the/a derived coend. |
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670 Or maybe that $F_*(M)$ is quasi-isomorphic (or perhaps homotopic) to the Hochschild |
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671 complex of $M$.} |
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672 This follows from the following lemmas: |
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673 \begin{itemize} |
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674 \item $F_*(M_1 \oplus M_2) \cong F_*(M_1) \oplus F_*(M_2)$. |
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675 \item An exact sequence $0 \to M_1 \to M_2 \to M_3 \to 0$ |
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676 gives rise to an exact sequence $0 \to F_*(M_1) \to F_*(M_2) \to F_*(M_3) \to 0$. |
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677 (See below for proof.) |
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678 \item $F_*(C\otimes C)$ (the free $C$-$C$-bimodule with one generator) is |
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679 quasi-isomorphic to the 0-step complex $C$. |
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680 (See below for proof.) |
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681 \item $F_*(C)$ (here $C$ is wearing its $C$-$C$-bimodule hat) is quasi-isomorphic to $\bc_*(S^1)$. |
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682 (See below for proof.) |
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683 \end{itemize} |
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684 |
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685 First we show that $F_*(C\otimes C)$ is |
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686 quasi-isomorphic to the 0-step complex $C$. |
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687 |
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688 Let $F'_* \sub F_*(C\otimes C)$ be the subcomplex where the label of |
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689 the point $*$ is $1 \otimes 1 \in C\otimes C$. |
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690 We will show that the inclusion $i: F'_* \to F_*(C\otimes C)$ is a quasi-isomorphism. |
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691 |
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692 Fix a small $\ep > 0$. |
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693 Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$. |
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694 Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex |
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695 generated by blob diagrams $b$ such that $B_\ep$ is either disjoint from |
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696 or contained in each blob of $b$, and the two boundary points of $B_\ep$ are not labeled points of $b$. |
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697 For a field (picture) $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$ |
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698 labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$. |
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699 (See Figure xxxx.) |
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700 Note that $y - s_\ep(y) \in U(B_\ep)$. |
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701 \nn{maybe it's simpler to assume that there are no labeled points, other than $*$, in $B_\ep$.} |
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702 |
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703 Define a degree 1 chain map $j_\ep : F^\ep_* \to F^\ep_*$ as follows. |
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704 Let $x \in F^\ep_*$ be a blob diagram. |
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705 If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $B_\ep$ to |
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706 $x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$. |
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707 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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708 Let $y_i$ be the restriction of $z_i$ to $B_\ep$. |
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709 Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$, |
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710 and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$. |
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711 Define $j_\ep(x) = \sum x_i$. |
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712 \nn{need to check signs coming from blob complex differential} |
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713 |
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714 Note that if $x \in F'_* \cap F^\ep_*$ then $j_\ep(x) \in F'_*$ also. |
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715 |
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716 The key property of $j_\ep$ is |
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717 \eq{ |
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718 \bd j_\ep + j_\ep \bd = \id - \sigma_\ep , |
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719 } |
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720 where $\sigma_\ep : F^\ep_* \to F^\ep_*$ is given by replacing the restriction $y$ of each field |
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721 mentioned in $x \in F^\ep_*$ with $s_\ep(y)$. |
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722 Note that $\sigma_\ep(x) \in F'_*$. |
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723 |
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724 If $j_\ep$ were defined on all of $F_*(C\otimes C)$, it would show that $\sigma_\ep$ |
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725 is a homotopy inverse to the inclusion $F'_* \to F_*(C\otimes C)$. |
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726 One strategy would be to try to stitch together various $j_\ep$ for progressively smaller |
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727 $\ep$ and show that $F'_*$ is homotopy equivalent to $F_*(C\otimes C)$. |
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728 Instead, we'll be less ambitious and just show that |
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729 $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$. |
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730 |
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731 If $x$ is a cycle in $F_*(C\otimes C)$, then for sufficiently small $\ep$ we have |
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732 $x \in F_*^\ep$. |
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733 (This is true for any chain in $F_*(C\otimes C)$, since chains are sums of |
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734 finitely many blob diagrams.) |
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735 Then $x$ is homologous to $s_\ep(x)$, which is in $F'_*$, so the inclusion map |
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736 $F'_* \sub F_*(C\otimes C)$ is surjective on homology. |
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737 If $y \in F_*(C\otimes C)$ and $\bd y = x \in F'_*$, then $y \in F^\ep_*$ for some $\ep$ |
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738 and |
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739 \eq{ |
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740 \bd y = \bd (\sigma_\ep(y) + j_\ep(x)) . |
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741 } |
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742 Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology. |
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743 This completes the proof that $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$. |
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744 |
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745 \medskip |
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746 |
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747 Let $F''_* \sub F'_*$ be the subcomplex of $F'_*$ where $*$ is not contained in any blob. |
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748 We will show that the inclusion $i: F''_* \to F'_*$ is a homotopy equivalence. |
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749 |
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750 First, a lemma: Let $G''_*$ and $G'_*$ be defined the same as $F''_*$ and $F'_*$, except with |
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751 $S^1$ replaced some (any) neighborhood of $* \in S^1$. |
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752 Then $G''_*$ and $G'_*$ are both contractible |
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753 and the inclusion $G''_* \sub G'_*$ is a homotopy equivalence. |
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754 For $G'_*$ the proof is the same as in (\ref{bcontract}), except that the splitting |
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755 $G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$. |
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756 For $G''_*$ we note that any cycle is supported \nn{need to establish terminology for this; maybe |
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757 in ``basic properties" section above} away from $*$. |
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758 Thus any cycle lies in the image of the normal blob complex of a disjoint union |
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759 of two intervals, which is contractible by (\ref{bcontract}) and (\ref{disjunion}). |
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760 Actually, we need the further (easy) result that the inclusion |
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761 $G''_* \to G'_*$ induces an isomorphism on $H_0$. |
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762 |
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763 Next we construct a degree 1 map (homotopy) $h: F'_* \to F'_*$ such that |
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764 for all $x \in F'_*$ we have |
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765 \eq{ |
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766 x - \bd h(x) - h(\bd x) \in F''_* . |
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767 } |
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768 Since $F'_0 = F''_0$, we can take $h_0 = 0$. |
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769 Let $x \in F'_1$, with single blob $B \sub S^1$. |
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770 If $* \notin B$, then $x \in F''_1$ and we define $h_1(x) = 0$. |
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771 If $* \in B$, then we work in the image of $G'_*$ and $G''_*$ (with respect to $B$). |
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772 Choose $x'' \in G''_1$ such that $\bd x'' = \bd x$. |
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773 Since $G'_*$ is contractible, there exists $y \in G'_2$ such that $\bd y = x - x''$. |
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774 Define $h_1(x) = y$. |
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775 The general case is similar, except that we have to take lower order homotopies into account. |
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776 Let $x \in F'_k$. |
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777 If $*$ is not contained in any of the blobs of $x$, then define $h_k(x) = 0$. |
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778 Otherwise, let $B$ be the outermost blob of $x$ containing $*$. |
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779 By xxxx above, $x = x' \bullet p$, where $x'$ is supported on $B$ and $p$ is supported away from $B$. |
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780 So $x' \in G'_l$ for some $l \le k$. |
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781 Choose $x'' \in G''_l$ such that $\bd x'' = \bd (x' - h_{l-1}\bd x')$. |
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782 Choose $y \in G'_{l+1}$ such that $\bd y = x' - x'' - h_{l-1}\bd x'$. |
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783 Define $h_k(x) = y \bullet p$. |
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784 This completes the proof that $i: F''_* \to F'_*$ is a homotopy equivalence. |
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785 \nn{need to say above more clearly and settle on notation/terminology} |
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786 |
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787 Finally, we show that $F''_*$ is contractible. |
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788 \nn{need to also show that $H_0$ is the right thing; easy, but I won't do it now} |
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789 Let $x$ be a cycle in $F''_*$. |
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790 The union of the supports of the diagrams in $x$ does not contain $*$, so there exists a |
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791 ball $B \subset S^1$ containing the union of the supports and not containing $*$. |
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792 Adding $B$ as a blob to $x$ gives a contraction. |
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793 \nn{need to say something else in degree zero} |
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794 |
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795 This completes the proof that $F_*(C\otimes C)$ is |
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796 homotopic to the 0-step complex $C$. |
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797 |
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798 \medskip |
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799 |
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800 Next we show that $F_*(C)$ is quasi-isomorphic to $\bc_*(S^1)$. |
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801 $F_*(C)$ differs from $\bc_*(S^1)$ only in that the base point * |
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802 is always a labeled point in $F_*(C)$, while in $\bc_*(S^1)$ it may or may not be. |
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803 In other words, there is an inclusion map $i: F_*(C) \to \bc_*(S^1)$. |
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804 |
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805 We define a quasi-inverse \nn{right term?} $s: \bc_*(S^1) \to F_*(C)$ to the inclusion as follows. |
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806 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
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807 * is a labeled point in $y$. |
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808 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
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809 Let $x \in \bc_*(S^1)$. |
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810 Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in |
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811 $x$ with $y$. |
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812 It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
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813 |
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814 Let $G^\ep_* \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points |
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815 in a neighborhood $B_\ep$ of *, except perhaps *. |
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816 Note that for any chain $x \in \bc_*(S^1)$, $x \in G^\ep_*$ for sufficiently small $\ep$. |
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817 \nn{rest of argument goes similarly to above} |
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818 |
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819 \bigskip |
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820 |
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821 \nn{still need to prove exactness claim} |
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822 |
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823 \nn{What else needs to be said to establish quasi-isomorphism to Hochschild complex? |
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824 Do we need a map from hoch to blob? |
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825 Does the above exactness and contractibility guarantee such a map without writing it |
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826 down explicitly? |
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827 Probably it's worth writing down an explicit map even if we don't need to.} |
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828 |
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829 |
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830 We can also describe explicitly a map from the standard Hochschild |
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831 complex to the blob complex on the circle. \nn{What properties does this |
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832 map have?} |
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833 |
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834 \begin{figure}% |
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835 $$\mathfig{0.6}{barycentric/barycentric}$$ |
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836 \caption{The Hochschild chain $a \tensor b \tensor c$ is sent to |
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837 the sum of six blob $2$-chains, corresponding to a barycentric subdivision of a $2$-simplex.} |
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838 \label{fig:Hochschild-example}% |
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839 \end{figure} |
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840 |
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841 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. |
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842 The edges marked $x, y$ and $z$ carry the $1$-chains |
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843 \begin{align*} |
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844 x & = \mathfig{0.1}{barycentric/ux} & u_x = \mathfig{0.1}{barycentric/ux_ca} - \mathfig{0.1}{barycentric/ux_c-a} \\ |
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845 y & = \mathfig{0.1}{barycentric/uy} & u_y = \mathfig{0.1}{barycentric/uy_cab} - \mathfig{0.1}{barycentric/uy_ca-b} \\ |
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846 z & = \mathfig{0.1}{barycentric/uz} & u_z = \mathfig{0.1}{barycentric/uz_c-a-b} - \mathfig{0.1}{barycentric/uz_cab} |
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847 \end{align*} |
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848 and the $2$-chain labelled $A$ is |
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849 \begin{equation*} |
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850 A = \mathfig{0.1}{barycentric/Ax}+\mathfig{0.1}{barycentric/Ay}. |
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851 \end{equation*} |
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852 Note that we then have |
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853 \begin{equation*} |
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854 \bdy A = x+y+z. |
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855 \end{equation*} |
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856 |
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857 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, |
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858 $$\phi\left(\Tensor_{i=1}^n a_i\right) = \sum_{\pi} \phi^\pi(a_1, \ldots, a_n)$$ |
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859 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!) |
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860 |
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861 |
583 |
862 \section{Action of $C_*(\Diff(X))$} \label{diffsect} |
584 \section{Action of $C_*(\Diff(X))$} \label{diffsect} |
863 |
585 |
864 Let $CD_*(X)$ denote $C_*(\Diff(X))$, the singular chain complex of |
586 Let $CD_*(X)$ denote $C_*(\Diff(X))$, the singular chain complex of |
865 the space of diffeomorphisms |
587 the space of diffeomorphisms |