874 |
874 |
875 \section{Hochschild homology when $n=1$} |
875 \section{Hochschild homology when $n=1$} |
876 \label{sec:hochschild} |
876 \label{sec:hochschild} |
877 \input{text/hochschild} |
877 \input{text/hochschild} |
878 |
878 |
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879 |
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880 |
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881 |
879 \section{Action of $\CD{X}$} |
882 \section{Action of $\CD{X}$} |
880 \label{sec:evaluation} |
883 \label{sec:evaluation} |
881 |
884 \input{text/evmap} |
882 Let $CD_*(X, Y)$ denote $C_*(\Diff(X \to Y))$, the singular chain complex of |
885 |
883 the space of diffeomorphisms |
886 |
884 \nn{or homeomorphisms} |
887 |
885 between the $n$-manifolds $X$ and $Y$ (extending a fixed diffeomorphism $\bd X \to \bd Y$). |
888 |
886 For convenience, we will permit the singular cells generating $CD_*(X, Y)$ to be more general |
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887 than simplices --- they can be based on any linear polyhedron. |
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888 \nn{be more restrictive here? does more need to be said?} |
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889 We also will use the abbreviated notation $CD_*(X) \deq CD_*(X, X)$. |
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890 |
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891 \begin{prop} \label{CDprop} |
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892 For each $n$-manifold $X$ there is a chain map |
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893 \eq{ |
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894 e_X : CD_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) . |
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895 } |
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896 On $CD_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X, Y)$ on $\bc_*(X)$ |
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897 (Proposition (\ref{diff0prop})). |
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898 For any splittings $X = X_1 \cup X_2$ and $Y = Y_1 \cup Y_2$, |
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899 the following diagram commutes up to homotopy |
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900 \eq{ \xymatrix{ |
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901 CD_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_X} & \bc_*(Y) \\ |
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902 CD_*(X_1, Y_1) \otimes CD_*(X_2, Y_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
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903 \ar@/_4ex/[r]_{e_{X_1} \otimes e_{X_2}} \ar[u]^{\gl \otimes \gl} & |
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904 \bc_*(Y_1) \otimes \bc_*(Y_2) \ar[u]_{\gl} |
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905 } } |
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906 Any other map satisfying the above two properties is homotopic to $e_X$. |
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907 \end{prop} |
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908 |
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909 \nn{need to rewrite for self-gluing instead of gluing two pieces together} |
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910 |
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911 \nn{Should say something stronger about uniqueness. |
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912 Something like: there is |
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913 a contractible subcomplex of the complex of chain maps |
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914 $CD_*(X) \otimes \bc_*(X) \to \bc_*(X)$ (0-cells are the maps, 1-cells are homotopies, etc.), |
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915 and all choices in the construction lie in the 0-cells of this |
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916 contractible subcomplex. |
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917 Or maybe better to say any two choices are homotopic, and |
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918 any two homotopies and second order homotopic, and so on.} |
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919 |
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920 \nn{Also need to say something about associativity. |
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921 Put it in the above prop or make it a separate prop? |
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922 I lean toward the latter.} |
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923 \medskip |
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924 |
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925 The proof will occupy the remainder of this section. |
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926 \nn{unless we put associativity prop at end} |
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927 |
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928 Without loss of generality, we will assume $X = Y$. |
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929 |
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930 \medskip |
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931 |
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932 Let $f: P \times X \to X$ be a family of diffeomorphisms and $S \sub X$. |
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933 We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all |
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934 $x \notin S$ and $p, q \in P$. Equivalently, $f$ is supported on $S$ if there is a family of diffeomorphisms $f' : P \times S \to S$ and a `background' |
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935 diffeomorphism $f_0 : X \to X$ so that |
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936 \begin{align} |
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937 f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\ |
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938 \intertext{and} |
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939 f(p,x) & = f_0(x) \;\;\;\; \mbox{for}\; (p, x) \in {P \times (X \setmin S)}. |
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940 \end{align} |
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941 Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$. |
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942 |
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943 Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
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944 A $k$-parameter family of diffeomorphisms $f: P \times X \to X$ is |
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945 {\it adapted to $\cU$} if there is a factorization |
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946 \eq{ |
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947 P = P_1 \times \cdots \times P_m |
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948 } |
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949 (for some $m \le k$) |
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950 and families of diffeomorphisms |
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951 \eq{ |
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952 f_i : P_i \times X \to X |
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953 } |
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954 such that |
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955 \begin{itemize} |
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956 \item each $f_i$ is supported on some connected $V_i \sub X$; |
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957 \item the sets $V_i$ are mutually disjoint; |
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958 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
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959 where $k_i = \dim(P_i)$; and |
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960 \item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$ |
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961 for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$. |
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962 \end{itemize} |
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963 A chain $x \in CD_k(X)$ is (by definition) adapted to $\cU$ if it is the sum |
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964 of singular cells, each of which is adapted to $\cU$. |
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965 |
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966 (Actually, in this section we will only need families of diffeomorphisms to be |
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967 {\it weakly adapted} to $\cU$, meaning that the support of $f$ is contained in the union |
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968 of at most $k$ of the $U_\alpha$'s.) |
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969 |
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970 \begin{lemma} \label{extension_lemma} |
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971 Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
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972 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
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973 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
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974 \end{lemma} |
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975 |
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976 The proof will be given in Section \ref{sec:localising}. |
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977 |
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978 \medskip |
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979 |
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980 Before diving into the details, we outline our strategy for the proof of Proposition \ref{CDprop}. |
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981 |
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982 Let $p$ be a singular cell in $CD_*(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
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983 Suppose that there exists $V \sub X$ such that |
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984 \begin{enumerate} |
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985 \item $V$ is homeomorphic to a disjoint union of balls, and |
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986 \item $\supp(p) \cup \supp(b) \sub V$. |
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987 \end{enumerate} |
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988 |
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989 \medskip |
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990 |
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991 \nn{to be continued....} |
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992 |
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993 |
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994 %\nn{say something about associativity here} |
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995 |
889 |
996 \input{text/A-infty.tex} |
890 \input{text/A-infty.tex} |
997 |
891 |
998 \input{text/gluing.tex} |
892 \input{text/gluing.tex} |
999 |
893 |