877 \input{text/hochschild} |
877 \input{text/hochschild} |
878 |
878 |
879 \section{Action of $\CD{X}$} |
879 \section{Action of $\CD{X}$} |
880 \label{sec:evaluation} |
880 \label{sec:evaluation} |
881 |
881 |
882 Let $CD_*(X)$ denote $C_*(\Diff(X))$, the singular chain complex of |
882 Let $CD_*(X, Y)$ denote $C_*(\Diff(X \to Y))$, the singular chain complex of |
883 the space of diffeomorphisms |
883 the space of diffeomorphisms |
884 of the $n$-manifold $X$ (fixed on $\bd X$). |
884 \nn{or homeomorphisms} |
885 For convenience, we will permit the singular cells generating $CD_*(X)$ to be more general |
885 between the $n$-manifolds $X$ and $Y$ (extending a fixed diffeomorphism $\bd X \to \bd Y$). |
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886 For convenience, we will permit the singular cells generating $CD_*(X, Y)$ to be more general |
886 than simplices --- they can be based on any linear polyhedron. |
887 than simplices --- they can be based on any linear polyhedron. |
887 \nn{be more restrictive here? does more need to be said?} |
888 \nn{be more restrictive here? does more need to be said?} |
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889 We also will use the abbreviated notation $CD_*(X) \deq CD_*(X, X)$. |
888 |
890 |
889 \begin{prop} \label{CDprop} |
891 \begin{prop} \label{CDprop} |
890 For each $n$-manifold $X$ there is a chain map |
892 For each $n$-manifold $X$ there is a chain map |
891 \eq{ |
893 \eq{ |
892 e_X : CD_*(X) \otimes \bc_*(X) \to \bc_*(X) . |
894 e_X : CD_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) . |
893 } |
895 } |
894 On $CD_0(X) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X)$ on $\bc_*(X)$ |
896 On $CD_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X, Y)$ on $\bc_*(X)$ |
895 (Proposition (\ref{diff0prop})). |
897 (Proposition (\ref{diff0prop})). |
896 For any splitting $X = X_1 \cup X_2$, the following diagram commutes |
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 |
897 \eq{ \xymatrix{ |
900 \eq{ \xymatrix{ |
898 CD_*(X) \otimes \bc_*(X) \ar[r]^{e_X} & \bc_*(X) \\ |
901 CD_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_X} & \bc_*(Y) \\ |
899 CD_*(X_1) \otimes CD_*(X_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
902 CD_*(X_1, Y_1) \otimes CD_*(X_2, Y_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
900 \ar@/_4ex/[r]_{e_{X_1} \otimes e_{X_2}} \ar[u]^{\gl \otimes \gl} & |
903 \ar@/_4ex/[r]_{e_{X_1} \otimes e_{X_2}} \ar[u]^{\gl \otimes \gl} & |
901 \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl} |
904 \bc_*(Y_1) \otimes \bc_*(Y_2) \ar[u]_{\gl} |
902 } } |
905 } } |
903 Any other map satisfying the above two properties is homotopic to $e_X$. |
906 Any other map satisfying the above two properties is homotopic to $e_X$. |
904 \end{prop} |
907 \end{prop} |
905 |
908 |
906 \nn{need to rewrite for self-gluing instead of gluing two pieces together} |
909 \nn{need to rewrite for self-gluing instead of gluing two pieces together} |
953 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
958 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
954 where $k_i = \dim(P_i)$; and |
959 where $k_i = \dim(P_i)$; and |
955 \item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$ |
960 \item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$ |
956 for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$. |
961 for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$. |
957 \end{itemize} |
962 \end{itemize} |
958 A chain $x \in C_k(\Diff(X))$ is (by definition) adapted to $\cU$ if it is the sum |
963 A chain $x \in CD_k(X)$ is (by definition) adapted to $\cU$ if it is the sum |
959 of singular cells, each of which is adapted to $\cU$. |
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.) |
960 |
969 |
961 \begin{lemma} \label{extension_lemma} |
970 \begin{lemma} \label{extension_lemma} |
962 Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
971 Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
963 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
972 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
964 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
973 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
966 |
975 |
967 The proof will be given in Section \ref{sec:localising}. |
976 The proof will be given in Section \ref{sec:localising}. |
968 |
977 |
969 \medskip |
978 \medskip |
970 |
979 |
971 The strategy for the proof of Proposition \ref{CDprop} is as follows. |
980 Before diving into the details, we outline our strategy for the proof of Proposition \ref{CDprop}. |
972 We will identify a subcomplex |
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973 \[ |
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974 G_* \sub CD_*(X) \otimes \bc_*(X) |
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975 \] |
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976 on which the evaluation map is uniquely determined (up to homotopy) by the conditions |
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977 in \ref{CDprop}. |
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978 We then show that the inclusion of $G_*$ into the full complex |
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979 is an equivalence in the appropriate sense. |
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980 \nn{need to be more specific here} |
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981 |
981 |
982 Let $p$ be a singular cell in $CD_*(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
982 Let $p$ be a singular cell in $CD_*(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
983 Roughly speaking, $p\otimes b$ is in $G_*$ if each component $V$ of the support of $p$ |
983 Suppose that there exists $V \sub X$ such that |
984 intersects at most one blob $B$ of $b$. |
984 \begin{enumerate} |
985 Since $V \cup B$ might not itself be a ball, we need a more careful and complicated definition. |
985 \item $V$ is homeomorphic to a disjoint union of balls, and |
986 Choose a metric for $X$. |
986 \item $\supp(p) \cup \supp(b) \sub V$. |
987 We define $p\otimes b$ to be in $G_*$ if there exist $\epsilon > 0$ such that |
987 \end{enumerate} |
988 $\supp(p) \cup N_\epsilon(b)$ is a union of balls, where $N_\epsilon(b)$ denotes the epsilon |
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989 neighborhood of the support of $b$. |
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990 \nn{maybe also require that $N_\delta(b)$ is a union of balls for all $\delta<\epsilon$.} |
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991 |
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992 \nn{need to worry about case where the intrinsic support of $p$ is not a union of balls. |
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993 probably we can just stipulate that it is (i.e. only consider families of diffeos with this property). |
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994 maybe we should build into the definition of ``adapted" that support takes up all of $U_i$.} |
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995 |
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996 \nn{need to eventually show independence of choice of metric. maybe there's a better way than |
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997 choosing a metric. perhaps just choose a nbd of each ball, but I think I see problems |
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998 with that as well. |
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999 the bottom line is that we need a scheme for choosing unions of balls |
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1000 which satisfies the $C$, $C'$, $C''$ claim made a few paragraphs below.} |
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1001 |
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1002 Next we define the evaluation map $e_X$ on $G_*$. |
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1003 We'll proceed inductively on $G_i$. |
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1004 The induction starts on $G_0$, where the evaluation map is determined |
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1005 by the action of $\Diff(X)$ on $\bc_*(X)$ |
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1006 because $G_0 \sub CD_0\otimes \bc_0$. |
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1007 Assume we have defined the evaluation map up to $G_{k-1}$ and |
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1008 let $p\otimes b$ be a generator of $G_k$. |
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1009 Let $C \sub X$ be a union of balls (as described above) containing $\supp(p)\cup\supp(b)$. |
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1010 There is a factorization $p = g \circ p'$, where $g\in \Diff(X)$ and $p'$ is a family of diffeomorphisms which is the identity outside of $C$. |
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1011 Let $b = b'\bullet b''$, where $b' \in \bc_*(C)$ and $b'' \in \bc_0(X\setmin C)$. |
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1012 We may assume inductively |
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1013 (cf the end of this paragraph) |
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1014 that $e_X(\bd(p\otimes b))$ has a similar factorization $x\bullet g(b'')$, where |
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1015 $x \in \bc_*(g(C))$ and $\bd x = 0$. |
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1016 Since $\bc_*(g(C))$ is contractible, there exists $y \in \bc_*(g(C))$ such that $\bd y = x$. |
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1017 Define $e_X(p\otimes b) = y\bullet g(b'')$. |
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1018 |
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1019 We now show that $e_X$ on $G_*$ is, up to homotopy, independent of the various choices made. |
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1020 If we make a different series of choice of the chain $y$ in the previous paragraph, |
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1021 we can inductively construct a homotopy between the two sets of choices, |
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1022 again relying on the contractibility of $\bc_*(g(G))$. |
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1023 A similar argument shows that this homotopy is unique up to second order homotopy, and so on. |
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1024 |
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1025 Given a different set of choices $\{C'\}$ of the unions of balls $\{C\}$, |
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1026 we can find a third set of choices $\{C''\}$ such that $C, C' \sub C''$. |
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1027 The argument now proceeds as in the previous paragraph. |
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1028 \nn{should maybe say more here; also need to back up claim about third set of choices} |
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1029 \nn{this definitely needs reworking} |
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1030 |
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1031 Next we show that given $x \in CD_*(X) \otimes \bc_*(X)$ with $\bd x \in G_*$, there exists |
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1032 a homotopy (rel $\bd x$) to $x' \in G_*$, and further that $x'$ and |
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1033 this homotopy are unique up to iterated homotopy. |
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1034 |
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1035 Given $k>0$ and a blob diagram $b$, we say that a cover $\cU$ of $X$ is $k$-compatible with |
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1036 $b$ if, for any $\{U_1, \ldots, U_k\} \sub \cU$, the union |
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1037 $U_1\cup\cdots\cup U_k$ is a union of balls which satisfies the condition used to define $G_*$ above. |
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1038 It follows from Lemma \ref{extension_lemma} |
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1039 that if $\cU$ is $k$-compatible with $b$ and |
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1040 $p$ is a $k$-parameter family of diffeomorphisms which is adapted to $\cU$, then |
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1041 $p\otimes b \in G_*$. |
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1042 \nn{maybe emphasize this more; it's one of the main ideas in the proof} |
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1043 |
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1044 Let $k$ be the degree of $x$ and choose a cover $\cU$ of $X$ such that $\cU$ is |
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1045 $k$-compatible with each of the (finitely many) blob diagrams occurring in $x$. |
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1046 We will use Lemma \ref{extension_lemma} with respect to the cover $\cU$ to |
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1047 construct the homotopy to $G_*$. |
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1048 First we will construct a homotopy $h \in G_*$ from $\bd x$ to a cycle $z$ such that |
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1049 each family of diffeomorphisms $p$ occurring in $z$ is adapted to $\cU$. |
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1050 Then we will construct a homotopy (rel boundary) $r$ from $x + h$ to $y$ such that |
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1051 each family of diffeomorphisms $p$ occurring in $y$ is adapted to $\cU$. |
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1052 This implies that $y \in G_*$. |
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1053 The homotopy $r$ can also be thought of as a homotopy from $x$ to $y-h \in G_*$, and this is the homotopy we seek. |
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1054 |
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1055 We will define $h$ inductively on bidegrees $(0, k-1), (1, k-2), \ldots, (k-1, 0)$. |
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1056 Define $h$ to be zero on bidegree $(0, k-1)$. |
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1057 Let $p\otimes b$ be a generator occurring in $\bd x$ with bidegree $(1, k-2)$. |
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1058 Using Lemma \ref{extension_lemma}, construct a homotopy (rel $\bd$) $q$ from $p$ to $p'$ which is adapted to $\cU$. |
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1059 Define $h$ at $p\otimes b$ to be $q\otimes b$. |
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1060 Let $p'\otimes b'$ be a generator occurring in $\bd x$ with bidegree $(2, k-3)$. |
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1061 Let $s$ denote the sum of the $q$'s from the previous step for generators |
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1062 adjacent to $(\bd p')\otimes b'$. |
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1063 \nn{need to say more here} |
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1064 Apply Lemma \ref{extension_lemma} to $p'+s$ |
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1065 yielding a family of diffeos $q'$. |
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1066 Define $h$ at $p'\otimes b'$ to be $q'\otimes b'$. |
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1067 Continuing in this way, we define all of $h$. |
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1068 |
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1069 The homotopy $r$ is constructed similarly. |
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1070 |
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1071 \nn{need to say something about uniqueness of $r$, $h$ etc. |
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1072 postpone this until second draft.} |
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1073 |
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1074 At this point, we have finished defining the evaluation map. |
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1075 The uniqueness statement in the proposition is clear from the method of proof. |
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1076 All that remains is to show that the evaluation map gets along well with cutting and gluing, |
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1077 as claimed in the proposition. |
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1078 This is in fact not difficult, since the myriad choices involved in defining the |
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1079 evaluation map can be made in parallel for the top and bottom |
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1080 arrows in the commutative diagram. |
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1081 |
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1082 This completes the proof of Proposition \ref{CDprop}. |
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1083 |
988 |
1084 \medskip |
989 \medskip |
1085 |
990 |
1086 \nn{say something about associativity here} |
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} |
1087 |
995 |
1088 \input{text/A-infty.tex} |
996 \input{text/A-infty.tex} |
1089 |
997 |
1090 \input{text/gluing.tex} |
998 \input{text/gluing.tex} |
1091 |
999 |