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

   887 than simplices --- they can be based on any linear polyhedron.

   888 \nn{be more restrictive here?  does more need to be said?}

   889 We also will use the abbreviated notation $CD_*(X) \deq CD_*(X, X)$.

   890 

   891 \begin{prop}  \label{CDprop}

   892 For each $n$-manifold $X$ there is a chain map

   893 \eq{

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

   895 }

   896 On $CD_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X, Y)$ on $\bc_*(X)$

   897 (Proposition (\ref{diff0prop})).

   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

   900 \eq{ \xymatrix{

   901      CD_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_X}    & \bc_*(Y) \\

   902      CD_*(X_1, Y_1) \otimes CD_*(X_2, Y_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2)

   903         \ar@/_4ex/[r]_{e_{X_1} \otimes e_{X_2}}  \ar[u]^{\gl \otimes \gl}  &

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

   905 } }

   906 Any other map satisfying the above two properties is homotopic to $e_X$.

   907 \end{prop}

   908 

   909 \nn{need to rewrite for self-gluing instead of gluing two pieces together}

   910 

   911 \nn{Should say something stronger about uniqueness.

   912 Something like: there is

   913 a contractible subcomplex of the complex of chain maps

   914 $CD_*(X) \otimes \bc_*(X) \to \bc_*(X)$ (0-cells are the maps, 1-cells are homotopies, etc.),

   915 and all choices in the construction lie in the 0-cells of this

   916 contractible subcomplex.

   917 Or maybe better to say any two choices are homotopic, and

   918 any two homotopies and second order homotopic, and so on.}

   919 

   920 \nn{Also need to say something about associativity.

   921 Put it in the above prop or make it a separate prop?

   922 I lean toward the latter.}

   923 \medskip

   924 

   925 The proof will occupy the remainder of this section.

   926 \nn{unless we put associativity prop at end}

   927 

   928 Without loss of generality, we will assume $X = Y$.

   929 

   930 \medskip

   931 

   932 Let $f: P \times X \to X$ be a family of diffeomorphisms and $S \sub X$.

   933 We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all

   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'

   935 diffeomorphism $f_0 : X \to X$ so that

   936 \begin{align}

   937 	f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\

   938 \intertext{and}

   939 	f(p,x) & = f_0(x) \;\;\;\; \mbox{for}\; (p, x) \in {P \times (X \setmin S)}.

   940 \end{align}

   941 Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$.

   942 

   943 Let $\cU = \{U_\alpha\}$ be an open cover of $X$.

   944 A $k$-parameter family of diffeomorphisms $f: P \times X \to X$ is

   945 {\it adapted to $\cU$} if there is a factorization

   946 \eq{

   947     P = P_1 \times \cdots \times P_m

   948 }

   949 (for some $m \le k$)

   950 and families of diffeomorphisms

   951 \eq{

   952     f_i :  P_i \times X \to X

   953 }

   954 such that

   955 \begin{itemize}

   956 \item each $f_i$ is supported on some connected $V_i \sub X$;

   957 \item the sets $V_i$ are mutually disjoint;

   958 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s,

   959 where $k_i = \dim(P_i)$; and

   960 \item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$

   961 for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$.

   962 \end{itemize}

   963 A chain $x \in CD_k(X)$ is (by definition) adapted to $\cU$ if it is the sum

   964 of singular cells, each of which is adapted to $\cU$.

   965 

   966 (Actually, in this section we will only need families of diffeomorphisms to be 

   967 {\it weakly adapted} to $\cU$, meaning that the support of $f$ is contained in the union

   968 of at most $k$ of the $U_\alpha$'s.)

   969 

   970 \begin{lemma}  \label{extension_lemma}

   971 Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$.

   972 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$.

   973 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$.

   974 \end{lemma}

   975 

   976 The proof will be given in Section \ref{sec:localising}.

   977 

   978 \medskip

   979 

   980 Before diving into the details, we outline our strategy for the proof of Proposition \ref{CDprop}.

   981 

   982 Let $p$ be a singular cell in $CD_*(X)$ and $b$ be a blob diagram in $\bc_*(X)$.

   983 Suppose that there exists $V \sub X$ such that

   984 \begin{enumerate}

   985 \item $V$ is homeomorphic to a disjoint union of balls, and

   986 \item $\supp(p) \cup \supp(b) \sub V$.

   987 \end{enumerate}

   988 

   989 \medskip

   990 

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

   992 

   993 

   994 %\nn{say something about associativity here}