29 

    29 \long\def\noop#1{}

   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

   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 $\bd X \to \bd Y$).

   886 For convenience, we will permit the singular cells generating $CD_*(X, Y)$ to be more general

   892     e_X : CD_*(X) \otimes \bc_*(X) \to \bc_*(X) .

   894     e_X : CD_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) .

   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 $\bc_*(X)$

   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 $Y = Y_1 \cup Y_2$, 

   899 the following diagram commutes up to homotopy

   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)

   901             \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl}

   904             \bc_*(Y_1) \otimes \bc_*(Y_2) \ar[u]_{\gl}

   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

   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

   973 \[

   974     G_* \sub CD_*(X) \otimes \bc_*(X)

   975 \]

   976 on which the evaluation map is uniquely determined (up to homotopy) by the conditions

   977 in \ref{CDprop}.

   978 We then show that the inclusion of $G_*$ into the full complex

   979 is an equivalence in the appropriate sense.

   980 \nn{need to be more specific here}

   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

   989 neighborhood of the support of $b$.

   990 \nn{maybe also require that $N_\delta(b)$ is a union of balls for all $\delta<\epsilon$.}

   991 

   992 \nn{need to worry about case where the intrinsic support of $p$ is not a union of balls.

   993 probably we can just stipulate that it is (i.e. only consider families of diffeos with this property).

   994 maybe we should build into the definition of ``adapted" that support takes up all of $U_i$.}

   995 

   996 \nn{need to eventually show independence of choice of metric.  maybe there's a better way than

   997 choosing a metric.  perhaps just choose a nbd of each ball, but I think I see problems

   998 with that as well.

   999 the bottom line is that we need a scheme for choosing unions of balls

  1000 which satisfies the $C$, $C'$, $C''$ claim made a few paragraphs below.}

  1001 

  1002 Next we define the evaluation map $e_X$ on $G_*$.

  1003 We'll proceed inductively on $G_i$.

  1004 The induction starts on $G_0$, where the evaluation map is determined

  1005 by the action of $\Diff(X)$ on $\bc_*(X)$

  1006 because $G_0 \sub CD_0\otimes \bc_0$.

  1007 Assume we have defined the evaluation map up to $G_{k-1}$ and

  1008 let $p\otimes b$ be a generator of $G_k$.

  1009 Let $C \sub X$ be a union of balls (as described above) containing $\supp(p)\cup\supp(b)$.

  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 $C$.

  1011 Let $b = b'\bullet b''$, where $b' \in \bc_*(C)$ and $b'' \in \bc_0(X\setmin C)$.

  1012 We may assume inductively 

  1013 (cf the end of this paragraph)

  1014 that $e_X(\bd(p\otimes b))$ has a similar factorization $x\bullet g(b'')$, where

  1015 $x \in \bc_*(g(C))$ and $\bd x = 0$.

  1016 Since $\bc_*(g(C))$ is contractible, there exists $y \in \bc_*(g(C))$ such that $\bd y = x$.

  1017 Define $e_X(p\otimes b) = y\bullet g(b'')$.

  1018 

  1019 We now show that $e_X$ on $G_*$ is, up to homotopy, independent of the various choices made.

  1020 If we make a different series of choice of the chain $y$ in the previous paragraph,

  1021 we can inductively construct a homotopy between the two sets of choices,

  1022 again relying on the contractibility of $\bc_*(g(G))$.

  1023 A similar argument shows that this homotopy is unique up to second order homotopy, and so on.

  1024 

  1025 Given a different set of choices $\{C'\}$ of the unions of balls $\{C\}$,

  1026 we can find a third set of choices $\{C''\}$ such that $C, C' \sub C''$.

  1027 The argument now proceeds as in the previous paragraph.

  1028 \nn{should maybe say more here; also need to back up claim about third set of choices}

  1029 \nn{this definitely needs reworking}

  1030 

  1031 Next we show that given $x \in CD_*(X) \otimes \bc_*(X)$ with $\bd x \in G_*$, there exists

  1032 a homotopy (rel $\bd x$) to $x' \in G_*$, and further that $x'$ and

  1033 this homotopy are unique up to iterated homotopy.

  1034 

  1035 Given $k>0$ and a blob diagram $b$, we say that a cover $\cU$ of $X$ is $k$-compatible with

  1036 $b$ if, for any $\{U_1, \ldots, U_k\} \sub \cU$, the union

  1037 $U_1\cup\cdots\cup U_k$ is a union of balls which satisfies the condition used to define $G_*$ above.

  1038 It follows from Lemma \ref{extension_lemma}

  1039 that if $\cU$ is $k$-compatible with $b$ and

  1040 $p$ is a $k$-parameter family of diffeomorphisms which is adapted to $\cU$, then

  1041 $p\otimes b \in G_*$.

  1042 \nn{maybe emphasize this more; it's one of the main ideas in the proof}

  1043 

  1044 Let $k$ be the degree of $x$ and choose a cover $\cU$ of $X$ such that $\cU$ is

  1045 $k$-compatible with each of the (finitely many) blob diagrams occurring in $x$.

  1046 We will use Lemma \ref{extension_lemma} with respect to the cover $\cU$ to

  1047 construct the homotopy to $G_*$.

  1048 First we will construct a homotopy $h \in G_*$ from $\bd x$ to a cycle $z$ such that

  1049 each family of diffeomorphisms $p$ occurring in $z$ is adapted to $\cU$.

  1050 Then we will construct a homotopy (rel boundary) $r$ from $x + h$ to $y$ such that

  1051 each family of diffeomorphisms $p$ occurring in $y$ is adapted to $\cU$.

  1052 This implies that $y \in G_*$.

  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.

  1054 

  1055 We will define $h$ inductively on bidegrees $(0, k-1), (1, k-2), \ldots, (k-1, 0)$.

  1056 Define $h$ to be zero on bidegree $(0, k-1)$.

  1057 Let $p\otimes b$ be a generator occurring in $\bd x$ with bidegree $(1, k-2)$.

  1058 Using Lemma \ref{extension_lemma}, construct a homotopy (rel $\bd$) $q$ from $p$ to $p'$ which is adapted to $\cU$.

  1059 Define $h$ at $p\otimes b$ to be $q\otimes b$.

  1060 Let $p'\otimes b'$ be a generator occurring in $\bd x$ with bidegree $(2, k-3)$.

  1061 Let $s$ denote the sum of the $q$'s from the previous step for generators

  1062 adjacent to $(\bd p')\otimes b'$.

  1063 \nn{need to say more here}

  1064 Apply Lemma \ref{extension_lemma} to $p'+s$

  1065 yielding a family of diffeos $q'$.

  1066 Define $h$ at $p'\otimes b'$ to be $q'\otimes b'$.

  1067 Continuing in this way, we define all of $h$.

  1068 

  1069 The homotopy $r$ is constructed similarly.

  1070 

  1071 \nn{need to say something about uniqueness of $r$, $h$ etc.

  1072 postpone this until second draft.}

  1073 

  1074 At this point, we have finished defining the evaluation map.

  1075 The uniqueness statement in the proposition is clear from the method of proof.

  1076 All that remains is to show that the evaluation map gets along well with cutting and gluing,

  1077 as claimed in the proposition.

  1078 This is in fact not difficult, since the myriad choices involved in defining the

  1079 evaluation map can be made in parallel for the top and bottom

  1080 arrows in the commutative diagram.

  1081 

  1082 This completes the proof of Proposition \ref{CDprop}.

  1086 \nn{say something about associativity here}

   991 \nn{to be continued....}

   992 

   993 

   994 %\nn{say something about associativity here}