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%!TEX root = ../blob1.tex
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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Let $CD_*(X, Y)$ denote $C_*(\Diff(X \to Y))$, the singular chain complex of
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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the space of diffeomorphisms
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\nn{or homeomorphisms}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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between the $n$-manifolds $X$ and $Y$ (extending a fixed diffeomorphism $\bd X \to \bd Y$).
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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For convenience, we will permit the singular cells generating $CD_*(X, Y)$ to be more general
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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than simplices --- they can be based on any linear polyhedron.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\nn{be more restrictive here? does more need to be said?}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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We also will use the abbreviated notation $CD_*(X) \deq CD_*(X, X)$.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\begin{prop} \label{CDprop}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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For $n$-manifolds $X$ and $Y$ there is a chain map
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\eq{
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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e_{XY} : CD_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) .
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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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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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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(Proposition (\ref{diff0prop})).
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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For any splittings $X = X_1 \cup X_2$ and $Y = Y_1 \cup Y_2$,
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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the following diagram commutes up to homotopy
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\eq{ \xymatrix{
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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CD_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_{XY}} & \bc_*(Y) \\
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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CD_*(X_1, Y_1) \otimes CD_*(X_2, Y_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2)
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\ar@/_4ex/[r]_{e_{X_1Y_1} \otimes e_{X_2Y_2}} \ar[u]^{\gl \otimes \gl} &
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\bc_*(Y_1) \otimes \bc_*(Y_2) \ar[u]_{\gl}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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} }
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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Any other map satisfying the above two properties is homotopic to $e_X$.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\end{prop}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\nn{need to rewrite for self-gluing instead of gluing two pieces together}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\nn{Should say something stronger about uniqueness.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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Something like: there is
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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a contractible subcomplex of the complex of chain maps
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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$CD_*(X) \otimes \bc_*(X) \to \bc_*(X)$ (0-cells are the maps, 1-cells are homotopies, etc.),
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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and all choices in the construction lie in the 0-cells of this
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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contractible subcomplex.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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Or maybe better to say any two choices are homotopic, and
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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any two homotopies and second order homotopic, and so on.}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\nn{Also need to say something about associativity.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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Put it in the above prop or make it a separate prop?
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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I lean toward the latter.}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\medskip
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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The proof will occupy the remainder of this section.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\nn{unless we put associativity prop at end}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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Without loss of generality, we will assume $X = Y$.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\medskip
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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Let $f: P \times X \to X$ be a family of diffeomorphisms and $S \sub X$.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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$x \notin S$ 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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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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diffeomorphism $f_0 : X \to X$ so that
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\begin{align}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\intertext{and}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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f(p,x) & = f_0(x) \;\;\;\; \mbox{for}\; (p, x) \in {P \times (X \setmin S)}.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\end{align}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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Let $\cU = \{U_\alpha\}$ be an open cover of $X$.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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A $k$-parameter family of diffeomorphisms $f: P \times X \to X$ is
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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{\it adapted to $\cU$} if there is a factorization
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\eq{
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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P = P_1 \times \cdots \times P_m
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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(for some $m \le k$)
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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and families of diffeomorphisms
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\eq{
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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f_i : P_i \times X \to X
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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such that
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\begin{itemize}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\item each $f_i$ is supported on some connected $V_i \sub X$;
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\item the sets $V_i$ are mutually disjoint;
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s,
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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where $k_i = \dim(P_i)$; and
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\end{itemize}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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A chain $x \in CD_k(X)$ is (by definition) adapted to $\cU$ if it is the sum
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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of singular cells, each of which is adapted to $\cU$.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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(Actually, in this section we will only need families of diffeomorphisms to be
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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{\it weakly adapted} to $\cU$, meaning that the support of $f$ is contained in the union
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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of at most $k$ of the $U_\alpha$'s.)
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\begin{lemma} \label{extension_lemma}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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Furthermore, one can choose the homotopy so that its support is equal to the support of $x$.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\end{lemma}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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The proof will be given in Section \ref{sec:localising}.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\medskip
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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Before diving into the details, we outline our strategy for the proof of Proposition \ref{CDprop}.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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%Suppose for the moment that evaluation maps with the advertised properties exist.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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Let $p$ be a singular cell in $CD_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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Suppose that there exists $V \sub X$ such that
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\begin{enumerate}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\item $V$ is homeomorphic to a disjoint union of balls, and
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\item $\supp(p) \cup \supp(b) \sub V$.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\end{enumerate}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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Let $W = X \setmin V$, and let $V' = p(V)$ and $W' = p(W)$.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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We then have a factorization
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\[
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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p = \gl(q, r),
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\]
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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where $q \in CD_k(V, V')$ and $r' \in CD_0(W, W')$.
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We can also factorize $b = \gl(b_V, b_W)$, where $b_V\in \bc_*(V)$ and $b_W\in\bc_0(W)$.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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According to the commutative diagram of the proposition, we must have
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\[
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e_X(p\otimes b) = e_X(\gl(q\otimes b_V, r\otimes b_W)) =
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gl(e_{VV'}(q\otimes b_V), e_{WW'}(r\otimes b_W)) .
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\]
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Since $r$ is a plain, 0-parameter family of diffeomorphisms, we must have
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\[
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e_{WW'}(r\otimes b_W) = r(b_W),
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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125 |
\]
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126 |
where $r(b_W)$ denotes the obvious action of diffeomorphisms on blob diagrams (in
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this case a 0-blob diagram).
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Since $V'$ is a disjoint union of balls, $\bc_*(V')$ is acyclic in degrees $>0$
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(by \ref{disjunion} and \ref{bcontract}).
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Assuming inductively that we have already defined $e_{VV'}(\bd(q\otimes b_V))$,
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there is, up to (iterated) homotopy, a unique choice for $e_{VV'}(q\otimes b_V)$
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such that
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\[
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134 |
\bd(e_{VV'}(q\otimes b_V)) = e_{VV'}(\bd(q\otimes b_V)) .
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\]
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Thus the conditions of the proposition determine (up to homotopy) the evaluation
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map for generators $p\otimes b$ such that $\supp(p) \cup \supp(b)$ is contained in a disjoint
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union of balls.
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On the other hand, Lemma \ref{extension_lemma} allows us to homotope
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\nn{is this commonly used as a verb?} arbitrary generators to sums of generators with this property.
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\nn{should give a name to this property; also forward reference}
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This (roughly) establishes the uniqueness part of the proposition.
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To show existence, we must show that the various choices involved in constructing
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145 |
evaluation maps in this way affect the final answer only by a homotopy.
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\nn{maybe put a little more into the outline before diving into the details.}
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\nn{Note: At the moment this section is very inconsistent with respect to PL versus smooth,
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homeomorphism versus diffeomorphism, etc.
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We expect that everything is true in the PL category, but at the moment our proof
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avails itself to smooth techniques.
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Furthermore, it traditional in the literature to speak of $C_*(\Diff(X))$
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rather than $C_*(\Homeo(X))$.}
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\medskip
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Now for the details.
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Notation: Let $|b| = \supp(b)$, $|p| = \supp(p)$.
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Choose a metric on $X$.
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Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero
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(e.g.\ $\ep_i = 2^{-i}$).
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Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$
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converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$).
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Given a generator $p\otimes b$ of $CD_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $k$
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define
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\[
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N_{i,k}(p\ot b) \deq \Nbd_{k\ep_i}(|b|) \cup \Nbd_{k\delta_i}(|p|).
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\]
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In other words, we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized
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by $k$), with $\ep_i$ controlling the size of the buffer around $|b|$ and $\delta_i$ controlling
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the size of the buffer around $|p|$.
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Next we define subcomplexes $G_*^{i,m} \sub CD_*(X)\otimes \bc_*(X)$.
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Let $p\ot b$ be a generator of $CD_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b)
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= \deg(p) + \deg(b)$.
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$p\ot b$ is (by definition) in $G_*^{i,m}$ if either (a) $\deg(p) = 0$ or (b)
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there exist codimension-zero submanifolds $V_1,\ldots,V_m \sub X$ such that each $V_j$
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is homeomorphic to a disjoint union of balls and
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\[
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N_{i,k}(p\ot b) \subeq V_1 \subeq N_{i,k+1}(p\ot b)
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\subeq V_2 \subeq \cdots \subeq V_m \subeq N_{i,k+m}(p\ot b) .
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\]
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Further, we require (inductively) that $\bd(p\ot b) \in G_*^{i,m}$.
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We also require that $b$ is splitable (transverse) along the boundary of each $V_l$.
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Note that $G_*^{i,m+1} \subeq G_*^{i,m}$.
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As sketched above and explained in detail below,
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$G_*^{i,m}$ is a subcomplex where it is easy to define
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the evaluation map.
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The parameter $m$ controls the number of iterated homotopies we are able to construct.
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The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of
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$CD_*(X)\ot \bc_*(X)$.
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Next we define a chain map (dependent on some choices) $e: G_*^{i,m} \to \bc_*(X)$.
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Let $p\ot b \in G_*^{i,m}$.
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If $\deg(p) = 0$, define
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\[
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e(p\ot b) = p(b) ,
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\]
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where $p(b)$ denotes the obvious action of the diffeomorphism(s) $p$ on the blob diagram $b$.
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For general $p\ot b$ ($\deg(p) \ge 1$) assume inductively that we have already defined
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$e(p'\ot b')$ when $\deg(p') + \deg(b') < k = \deg(p) + \deg(b)$.
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Choose $V_1$ as above so that
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\[
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N_{i,k}(p\ot b) \subeq V_1 \subeq N_{i,k+1}(p\ot b) .
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\]
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Let $\bd(p\ot b) = \sum_j p_j\ot b_j$, and let $V_1^j$ be the choice of neighborhood
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of $|p_j|\cup |b_j|$ made at the preceding stage of the induction.
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For all $j$,
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\[
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V_1^j \subeq N_{i,(k-1)+1}(p_j\ot b_j) \subeq N_{i,k}(p\ot b) \subeq V_1 .
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\]
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(The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.)
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We therefore have splittings
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\[
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p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet f'' ,
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\]
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where $p' \in CD_*(V_1)$, $p'' \in CD_*(X\setmin V_1)$,
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$b' \in \bc_*(V_1)$, $b'' \in \bc_*(X\setmin V_1)$,
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$e' \in \bc_*(p(V_1))$, and $e'' \in \bc_*(p(X\setmin V_1))$.
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(Note that since the family of diffeomorphisms $p$ is constant (independent of parameters)
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near $\bd V_1)$, the expressions $p(V_1) \sub X$ and $p(X\setmin V_1) \sub X$ are
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unambiguous.)
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We also have that $\deg(b'') = 0 = \deg(p'')$.
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Choose $x' \in \bc_*(p(V_1))$ such that $\bd x' = f'$.
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This is possible by \nn{...}.
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Finally, define
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\[
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e(p\ot b) \deq x' \bullet p''(b'') .
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\]
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\medskip
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\nn{to be continued....}
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%\nn{say something about associativity here}
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