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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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Let $W = X \setmin V$, $W' = p(W)$ and $V' = X\setmin W'$.
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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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\[
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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p = \gl(q, r),
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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114 |
\]
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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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\[
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119 |
e_X(p\otimes b) = e_X(\gl(q\otimes b_V, r\otimes b_W)) =
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gl(e_{VV'}(q\otimes b_V), e_{WW'}(r\otimes b_W)) .
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\]
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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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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.
|
|
159 |
|
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160 |
Notation: Let $|b| = \supp(b)$, $|p| = \supp(p)$.
|
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161 |
|
|
162 |
Choose a metric on $X$.
|
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163 |
Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero
|
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164 |
(e.g.\ $\ep_i = 2^{-i}$).
|
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165 |
Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$
|
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166 |
converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$).
|
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|
167 |
Let $\phi_l$ be an increasing sequence of positive numbers
|
89
|
168 |
satisfying the inequalities of Lemma \ref{xx2phi}.
|
88
|
169 |
Given a generator $p\otimes b$ of $CD_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $l$
|
83
|
170 |
define
|
|
171 |
\[
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88
|
172 |
N_{i,l}(p\ot b) \deq \Nbd_{l\ep_i}(|b|) \cup \Nbd_{\phi_l\delta_i}(|p|).
|
83
|
173 |
\]
|
|
174 |
In other words, we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized
|
88
|
175 |
by $l$), with $\ep_i$ controlling the size of the buffers around $|b|$ and $\delta_i$ controlling
|
|
176 |
the size of the buffers around $|p|$.
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|
177 |
|
|
178 |
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 $k = \deg(p\ot b)
|
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180 |
= \deg(p) + \deg(b)$.
|
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181 |
$p\ot b$ is (by definition) in $G_*^{i,m}$ if either (a) $\deg(p) = 0$ or (b)
|
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|
182 |
there exist codimension-zero submanifolds $V_0,\ldots,V_m \sub X$ such that each $V_j$
|
83
|
183 |
is homeomorphic to a disjoint union of balls and
|
|
184 |
\[
|
84
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185 |
N_{i,k}(p\ot b) \subeq V_0 \subeq N_{i,k+1}(p\ot b)
|
|
186 |
\subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,k+m+1}(p\ot b) .
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83
|
187 |
\]
|
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188 |
Further, we require (inductively) that $\bd(p\ot b) \in G_*^{i,m}$.
|
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189 |
We also require that $b$ is splitable (transverse) along the boundary of each $V_l$.
|
|
190 |
|
|
191 |
Note that $G_*^{i,m+1} \subeq G_*^{i,m}$.
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73
|
192 |
|
83
|
193 |
As sketched above and explained in detail below,
|
|
194 |
$G_*^{i,m}$ is a subcomplex where it is easy to define
|
|
195 |
the evaluation map.
|
84
|
196 |
The parameter $m$ controls the number of iterated homotopies we are able to construct
|
87
|
197 |
(see Lemma \ref{m_order_hty}).
|
83
|
198 |
The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of
|
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|
199 |
$CD_*(X)\ot \bc_*(X)$ (see Lemma \ref{Gim_approx}).
|
83
|
200 |
|
|
201 |
Next we define a chain map (dependent on some choices) $e: G_*^{i,m} \to \bc_*(X)$.
|
|
202 |
Let $p\ot b \in G_*^{i,m}$.
|
|
203 |
If $\deg(p) = 0$, define
|
|
204 |
\[
|
|
205 |
e(p\ot b) = p(b) ,
|
|
206 |
\]
|
|
207 |
where $p(b)$ denotes the obvious action of the diffeomorphism(s) $p$ on the blob diagram $b$.
|
|
208 |
For general $p\ot b$ ($\deg(p) \ge 1$) assume inductively that we have already defined
|
|
209 |
$e(p'\ot b')$ when $\deg(p') + \deg(b') < k = \deg(p) + \deg(b)$.
|
84
|
210 |
Choose $V = V_0$ as above so that
|
83
|
211 |
\[
|
84
|
212 |
N_{i,k}(p\ot b) \subeq V \subeq N_{i,k+1}(p\ot b) .
|
83
|
213 |
\]
|
84
|
214 |
Let $\bd(p\ot b) = \sum_j p_j\ot b_j$, and let $V^j$ be the choice of neighborhood
|
83
|
215 |
of $|p_j|\cup |b_j|$ made at the preceding stage of the induction.
|
|
216 |
For all $j$,
|
|
217 |
\[
|
88
|
218 |
V^j \subeq N_{i,k}(p_j\ot b_j) \subeq N_{i,k}(p\ot b) \subeq V .
|
83
|
219 |
\]
|
|
220 |
(The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.)
|
|
221 |
We therefore have splittings
|
|
222 |
\[
|
86
|
223 |
p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet b'' ,
|
83
|
224 |
\]
|
84
|
225 |
where $p' \in CD_*(V)$, $p'' \in CD_*(X\setmin V)$,
|
|
226 |
$b' \in \bc_*(V)$, $b'' \in \bc_*(X\setmin V)$,
|
86
|
227 |
$f' \in \bc_*(p(V))$, and $f'' \in \bc_*(p(X\setmin V))$.
|
83
|
228 |
(Note that since the family of diffeomorphisms $p$ is constant (independent of parameters)
|
86
|
229 |
near $\bd V$, the expressions $p(V) \sub X$ and $p(X\setmin V) \sub X$ are
|
83
|
230 |
unambiguous.)
|
86
|
231 |
We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$.
|
|
232 |
%We also have that $\deg(b'') = 0 = \deg(p'')$.
|
84
|
233 |
Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$.
|
89
|
234 |
This is possible by \ref{bcontract}, \ref{disjunion} and \nn{property 2 of local relations (isotopy)}.
|
83
|
235 |
Finally, define
|
|
236 |
\[
|
|
237 |
e(p\ot b) \deq x' \bullet p''(b'') .
|
|
238 |
\]
|
73
|
239 |
|
84
|
240 |
Note that above we are essentially using the method of acyclic models.
|
|
241 |
For each generator $p\ot b$ we specify the acyclic (in positive degrees)
|
|
242 |
target complex $\bc_*(p(V)) \bullet p''(b'')$.
|
|
243 |
|
|
244 |
The definition of $e: G_*^{i,m} \to \bc_*(X)$ depends on two sets of choices:
|
|
245 |
The choice of neighborhoods $V$ and the choice of inverse boundaries $x'$.
|
88
|
246 |
The next lemma shows that up to (iterated) homotopy $e$ is independent
|
84
|
247 |
of these choices.
|
88
|
248 |
(Note that independence of choices of $x'$ (for fixed choices of $V$)
|
|
249 |
is a standard result in the method of acyclic models.)
|
84
|
250 |
|
88
|
251 |
%\begin{lemma}
|
|
252 |
%Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with
|
|
253 |
%different choices of $x'$ at each step.
|
|
254 |
%(Same choice of $V$ at each step.)
|
|
255 |
%Then $e$ and $\tilde{e}$ are homotopic via a homotopy in $\bc_*(p(V)) \bullet p''(b'')$.
|
|
256 |
%Any two choices of such a first-order homotopy are second-order homotopic, and so on,
|
|
257 |
%to arbitrary order.
|
|
258 |
%\end{lemma}
|
84
|
259 |
|
88
|
260 |
%\begin{proof}
|
|
261 |
%This is a standard result in the method of acyclic models.
|
|
262 |
%\nn{should we say more here?}
|
|
263 |
%\nn{maybe this lemma should be subsumed into the next lemma. probably it should.}
|
|
264 |
%\end{proof}
|
84
|
265 |
|
87
|
266 |
\begin{lemma} \label{m_order_hty}
|
84
|
267 |
Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with
|
|
268 |
different choices of $V$ (and hence also different choices of $x'$) at each step.
|
|
269 |
If $m \ge 1$ then $e$ and $\tilde{e}$ are homotopic.
|
|
270 |
If $m \ge 2$ then any two choices of this first-order homotopy are second-order homotopic.
|
|
271 |
And so on.
|
|
272 |
In other words, $e : G_*^{i,m} \to \bc_*(X)$ is well-defined up to $m$-th order homotopy.
|
|
273 |
\end{lemma}
|
|
274 |
|
|
275 |
\begin{proof}
|
|
276 |
We construct $h: G_*^{i,m} \to \bc_*(X)$ such that $\bd h + h\bd = e - \tilde{e}$.
|
|
277 |
$e$ and $\tilde{e}$ coincide on bidegrees $(0, j)$, so define $h$
|
|
278 |
to be zero there.
|
|
279 |
Assume inductively that $h$ has been defined for degrees less than $k$.
|
|
280 |
Let $p\ot b$ be a generator of degree $k$.
|
|
281 |
Choose $V_1$ as in the definition of $G_*^{i,m}$ so that
|
|
282 |
\[
|
|
283 |
N_{i,k+1}(p\ot b) \subeq V_1 \subeq N_{i,k+2}(p\ot b) .
|
|
284 |
\]
|
|
285 |
There are splittings
|
|
286 |
\[
|
|
287 |
p = p'_1\bullet p''_1 , \;\; b = b'_1\bullet b''_1 ,
|
|
288 |
\;\; e(p\ot b) - \tilde{e}(p\ot b) - h(\bd(p\ot b)) = f'_1\bullet f''_1 ,
|
|
289 |
\]
|
|
290 |
where $p'_1 \in CD_*(V_1)$, $p''_1 \in CD_*(X\setmin V_1)$,
|
|
291 |
$b'_1 \in \bc_*(V_1)$, $b''_1 \in \bc_*(X\setmin V_1)$,
|
|
292 |
$f'_1 \in \bc_*(p(V_1))$, and $f''_1 \in \bc_*(p(X\setmin V_1))$.
|
88
|
293 |
Inductively, $\bd f'_1 = 0$ and $f_1'' = p_1''(b_1'')$.
|
84
|
294 |
Choose $x'_1 \in \bc_*(p(V_1))$ so that $\bd x'_1 = f'_1$.
|
|
295 |
Define
|
|
296 |
\[
|
|
297 |
h(p\ot b) \deq x'_1 \bullet p''_1(b''_1) .
|
|
298 |
\]
|
|
299 |
This completes the construction of the first-order homotopy when $m \ge 1$.
|
|
300 |
|
|
301 |
The $j$-th order homotopy is constructed similarly, with $V_j$ replacing $V_1$ above.
|
|
302 |
\end{proof}
|
|
303 |
|
|
304 |
Note that on $G_*^{i,m+1} \subeq G_*^{i,m}$, we have defined two maps,
|
|
305 |
call them $e_{i,m}$ and $e_{i,m+1}$.
|
|
306 |
An easy variation on the above lemma shows that $e_{i,m}$ and $e_{i,m+1}$ are $m$-th
|
|
307 |
order homotopic.
|
|
308 |
|
85
|
309 |
Next we show how to homotope chains in $CD_*(X)\ot \bc_*(X)$ to one of the
|
|
310 |
$G_*^{i,m}$.
|
|
311 |
Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero.
|
|
312 |
Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$.
|
|
313 |
Let $h_j: CD_*(X)\to CD_*(X)$ be a chain map homotopic to the identity whose image is spanned by diffeomorphisms with support compatible with $\cU_j$, as described in Lemma \ref{xxxxx}.
|
86
|
314 |
Recall that $h_j$ and also the homotopy connecting it to the identity do not increase
|
85
|
315 |
supports.
|
|
316 |
Define
|
|
317 |
\[
|
|
318 |
g_j \deq h_j\circ h_{j-1} \circ \cdots \circ h_1 .
|
|
319 |
\]
|
|
320 |
The next lemma says that for all generators $p\ot b$ we can choose $j$ large enough so that
|
|
321 |
$g_j(p)\ot b$ lies in $G_*^{i,m}$, for arbitrary $m$ and sufficiently large $i$
|
|
322 |
(depending on $b$, $n = \deg(p)$ and $m$).
|
88
|
323 |
(Note: Don't confuse this $n$ with the top dimension $n$ used elsewhere in this paper.)
|
85
|
324 |
|
87
|
325 |
\begin{lemma} \label{Gim_approx}
|
85
|
326 |
Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CD_*(X)$.
|
|
327 |
Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$
|
|
328 |
there exists another constant $j_i$ such that for all $j \ge j_i$ and all $p\in CD_n(X)$
|
|
329 |
we have $g_j(p)\ot b \in G_*^{i,m}$.
|
|
330 |
\end{lemma}
|
|
331 |
|
|
332 |
\begin{proof}
|
|
333 |
Let $c$ be a subset of the blobs of $b$.
|
|
334 |
There exists $l > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < l$
|
|
335 |
and all such $c$.
|
86
|
336 |
(Here we are using a piecewise smoothness assumption for $\bd c$, and also
|
90
|
337 |
the fact that $\bd c$ is collared.
|
|
338 |
We need to consider all such $c$ because all generators appearing in
|
|
339 |
iterated boundaries of must be in $G_*^{i,m}$.)
|
85
|
340 |
|
86
|
341 |
Let $r = \deg(b)$ and
|
|
342 |
\[
|
90
|
343 |
t = r+n+m+1 = \deg(p\ot b) + m + 1.
|
86
|
344 |
\]
|
85
|
345 |
|
|
346 |
Choose $k = k_{bmn}$ such that
|
|
347 |
\[
|
86
|
348 |
t\ep_k < l
|
85
|
349 |
\]
|
|
350 |
and
|
|
351 |
\[
|
90
|
352 |
n\cdot (2 (\phi_t + 1) \delta_k) < \ep_k .
|
85
|
353 |
\]
|
|
354 |
Let $i \ge k_{bmn}$.
|
|
355 |
Choose $j = j_i$ so that
|
|
356 |
\[
|
90
|
357 |
\gamma_j < \delta_i
|
|
358 |
\]
|
|
359 |
and also so that $\phi_t \gamma_j$ is less than the constant $\rho(M)$ of Lemma \ref{xxzz11}.
|
|
360 |
|
|
361 |
Let $j \ge j_i$ and $p\in CD_n(X)$.
|
|
362 |
Let $q$ be a generator appearing in $g_j(p)$.
|
|
363 |
Note that $|q|$ is contained in a union of $n$ elements of the cover $\cU_j$,
|
|
364 |
which implies that $|q|$ is contained in a union of $n$ metric balls of radius $\delta_i$.
|
|
365 |
We must show that $q\ot b \in G_*^{i,m}$, which means finding neighborhoods
|
|
366 |
$V_0,\ldots,V_m \sub X$ of $|q|\cup |b|$ such that each $V_j$
|
|
367 |
is homeomorphic to a disjoint union of balls and
|
|
368 |
\[
|
|
369 |
N_{i,n}(q\ot b) \subeq V_0 \subeq N_{i,n+1}(q\ot b)
|
|
370 |
\subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,t}(q\ot b) .
|
|
371 |
\]
|
|
372 |
By repeated applications of Lemma \ref{xx2phi} we can find neighborhoods $U_0,\ldots,U_m$
|
|
373 |
of $|q|$, each homeomorphic to a disjoint union of balls, with
|
|
374 |
\[
|
|
375 |
\Nbd_{\phi_{n+l} \delta_i}(|q|) \subeq U_l \subeq \Nbd_{\phi_{n+l+1} \delta_i}(|q|) .
|
85
|
376 |
\]
|
90
|
377 |
The inequalities above \nn{give ref} guarantee that we can find $u_l$ with
|
|
378 |
\[
|
|
379 |
(n+l)\ep_i \le u_l \le (n+l+1)\ep_i
|
|
380 |
\]
|
|
381 |
such that each component of $U_l$ is either disjoint from $\Nbd_{u_l}(|b|)$ or contained in
|
|
382 |
$\Nbd_{u_l}(|b|)$.
|
|
383 |
This is because there are at most $n$ components of $U_l$, and each component
|
|
384 |
has radius $\le (\phi_t + 1) \delta_i$.
|
|
385 |
It follows that
|
|
386 |
\[
|
|
387 |
V_l \deq \Nbd_{u_l}(|b|) \cup U_l
|
|
388 |
\]
|
|
389 |
is homeomorphic to a disjoint union of balls and satisfies
|
|
390 |
\[
|
|
391 |
N_{i,n+l}(q\ot b) \subeq V_l \subeq N_{i,n+l+1}(q\ot b) .
|
|
392 |
\]
|
86
|
393 |
|
90
|
394 |
The same argument shows that each generator involved in iterated boundaries of $q\ot b$
|
|
395 |
is in $G_*^{i,m}$.
|
86
|
396 |
\end{proof}
|
|
397 |
|
|
398 |
In the next few lemmas we have made no effort to optimize the various bounds.
|
|
399 |
(The bounds are, however, optimal in the sense of minimizing the amount of work
|
|
400 |
we do. Equivalently, they are the first bounds we thought of.)
|
|
401 |
|
|
402 |
We say that a subset $S$ of a metric space has radius $\le r$ if $S$ is contained in
|
|
403 |
some metric ball of radius $r$.
|
|
404 |
|
|
405 |
\begin{lemma}
|
|
406 |
Let $S \sub \ebb^n$ (Euclidean $n$-space) have radius $\le r$.
|
|
407 |
Then $\Nbd_a(S)$ is homeomorphic to a ball for $a \ge 2r$.
|
|
408 |
\end{lemma}
|
|
409 |
|
|
410 |
\begin{proof} \label{xxyy2}
|
|
411 |
Let $S$ be contained in $B_r(y)$, $y \in \ebb^n$.
|
89
|
412 |
Note that if $a \ge 2r$ then $\Nbd_a(S) \sup B_r(y)$.
|
|
413 |
Let $z\in \Nbd_a(S) \setmin B_r(y)$.
|
|
414 |
Consider the triangle
|
|
415 |
\nn{give figure?} with vertices $z$, $y$ and $s$ with $s\in S$.
|
|
416 |
The length of the edge $yz$ is greater than $r$ which is greater
|
|
417 |
than the length of the edge $ys$.
|
|
418 |
It follows that the angle at $z$ is less than $\pi/2$ (less than $\pi/3$, in fact),
|
|
419 |
which means that points on the edge $yz$ near $z$ are closer to $s$ than $z$ is,
|
|
420 |
which implies that these points are also in $\Nbd_a(S)$.
|
|
421 |
Hence $\Nbd_a(S)$ is star-shaped with respect to $y$.
|
|
422 |
\end{proof}
|
|
423 |
|
|
424 |
If we replace $\ebb^n$ above with an arbitrary compact Riemannian manifold $M$,
|
|
425 |
the same result holds, so long as $a$ is not too large:
|
|
426 |
|
|
427 |
\begin{lemma} \label{xxzz11}
|
|
428 |
Let $M$ be a compact Riemannian manifold.
|
|
429 |
Then there is a constant $\rho(M)$ such that for all
|
|
430 |
subsets $S\sub M$ of radius $\le r$ and all $a$ such that $2r \le a \le \rho(M)$,
|
|
431 |
$\Nbd_a(S)$ is homeomorphic to a ball.
|
|
432 |
\end{lemma}
|
|
433 |
|
|
434 |
\begin{proof}
|
|
435 |
Choose $\rho = \rho(M)$ such that $3\rho/2$ is less than the radius of injectivity of $M$,
|
|
436 |
and also so that for any point $y\in M$ the geodesic coordinates of radius $3\rho/2$ around
|
|
437 |
$y$ distort angles by only a small amount.
|
|
438 |
Now the argument of the previous lemma works.
|
85
|
439 |
\end{proof}
|
|
440 |
|
|
441 |
|
89
|
442 |
|
|
443 |
\begin{lemma} \label{xx2phi}
|
|
444 |
Let $S \sub M$ be contained in a union (not necessarily disjoint)
|
86
|
445 |
of $k$ metric balls of radius $r$.
|
89
|
446 |
Let $\phi_1, \phi_2, \ldots$ be an increasing sequence of real numbers satisfying
|
|
447 |
$\phi_1 \ge 2$ and $\phi_{i+1} \ge \phi_i(2\phi_i + 2) + \phi_i$.
|
|
448 |
For convenience, let $\phi_0 = 0$.
|
|
449 |
Assume also that $\phi_k r \le \rho(M)$.
|
|
450 |
Then there exists a neighborhood $U$ of $S$,
|
|
451 |
homeomorphic to a disjoint union of balls, such that
|
86
|
452 |
\[
|
89
|
453 |
\Nbd_{\phi_{k-1} r}(S) \subeq U \subeq \Nbd_{\phi_k r}(S) .
|
86
|
454 |
\]
|
|
455 |
\end{lemma}
|
|
456 |
|
|
457 |
\begin{proof}
|
89
|
458 |
For $k=1$ this follows from Lemma \ref{xxzz11}.
|
|
459 |
Assume inductively that it holds for $k-1$.
|
86
|
460 |
Partition $S$ into $k$ disjoint subsets $S_1,\ldots,S_k$, each of radius $\le r$.
|
89
|
461 |
By Lemma \ref{xxzz11}, each $\Nbd_{\phi_{k-1} r}(S_i)$ is homeomorphic to a ball.
|
|
462 |
If these balls are disjoint, let $U$ be their union.
|
|
463 |
Otherwise, assume WLOG that $S_{k-1}$ and $S_k$ are distance less than $2\phi_{k-1}r$ apart.
|
|
464 |
Let $R_i = \Nbd_{\phi_{k-1} r}(S_i)$ for $i = 1,\ldots,k-2$
|
|
465 |
and $R_{k-1} = \Nbd_{\phi_{k-1} r}(S_{k-1})\cup \Nbd_{\phi_{k-1} r}(S_k)$.
|
|
466 |
Each $R_i$ is contained in a metric ball of radius $r' \deq (2\phi_{k-1}+2)r$.
|
91
|
467 |
Note that the defining inequality of the $\phi_i$ guarantees that
|
|
468 |
\[
|
|
469 |
\phi_{k-1}r' = \phi_{k-1}(2\phi_{k-1}+2)r \le \phi_k r \le \rho(M) .
|
|
470 |
\]
|
89
|
471 |
By induction, there is a neighborhood $U$ of $R \deq \bigcup_i R_i$,
|
|
472 |
homeomorphic to a disjoint union
|
|
473 |
of balls, and such that
|
86
|
474 |
\[
|
89
|
475 |
U \subeq \Nbd_{\phi_{k-1}r'}(R) = \Nbd_{t}(S) \subeq \Nbd_{\phi_k r}(S) ,
|
86
|
476 |
\]
|
89
|
477 |
where $t = \phi_{k-1}(2\phi_{k-1}+2)r + \phi_{k-1} r$.
|
86
|
478 |
\end{proof}
|
|
479 |
|
|
480 |
|
70
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
481 |
\medskip
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
482 |
|
86
|
483 |
|
92
|
484 |
\hrule\medskip\hrule\medskip
|
|
485 |
|
|
486 |
\nn{outline of what remains to be done:}
|
|
487 |
|
|
488 |
\begin{itemize}
|
|
489 |
\item We need to assemble the maps for the various $G^{i,m}$ into
|
|
490 |
a map for all of $CD_*\ot \bc_*$.
|
|
491 |
One idea: Think of the $g_j$ as a sort of homotopy (from $CD_*\ot \bc_*$ to itself)
|
|
492 |
parameterized by $[0,\infty)$. For each $p\ot b$ in $CD_*\ot \bc_*$ choose a sufficiently
|
|
493 |
large $j'$. Use these choices to reparameterize $g_\bullet$ so that each
|
|
494 |
$p\ot b$ gets pushed as far as the corresponding $j'$.
|
|
495 |
\item Independence of metric, $\ep_i$, $\delta_i$:
|
|
496 |
For a different metric etc. let $\hat{G}^{i,m}$ denote the alternate subcomplexes
|
|
497 |
and $\hat{N}_{i,l}$ the alternate neighborhoods.
|
|
498 |
Main idea is that for all $i$ there exists sufficiently large $k$ such that
|
|
499 |
$\hat{N}_{k,l} \sub N_{i,l}$, and similarly with the roles of $N$ and $\hat{N}$ reversed.
|
|
500 |
\item Also need to prove associativity.
|
|
501 |
\end{itemize}
|
86
|
502 |
|
|
503 |
|
92
|
504 |
\nn{to be continued....}
|
86
|
505 |
|
84
|
506 |
\noop{
|
|
507 |
|
|
508 |
\begin{lemma}
|
|
509 |
|
|
510 |
\end{lemma}
|
86
|
511 |
|
84
|
512 |
\begin{proof}
|
|
513 |
|
|
514 |
\end{proof}
|
|
515 |
|
|
516 |
}
|
|
517 |
|
|
518 |
|
70
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
519 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
520 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
521 |
%\nn{say something about associativity here}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
522 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
523 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
524 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
525 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
526 |
|