blob: comparison text/evmap.tex

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-:0488412c274b
+:3feb6e24a518
 %!TEX root = ../blob1.tex
-\section{Action of \texorpdfstring{$\CD{X}$}{$C_*(Diff(M))$}}
+\section{Action of \texorpdfstring{$\CH{X}$}{$C_*(Homeo(M))$}}
 \label{sec:evaluation}
-Let $CD_*(X, Y)$ denote $C_*(\Diff(X \to Y))$, the singular chain complex of
+Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of
-the space of diffeomorphisms
+the space of homeomorphisms
-\nn{or homeomorphisms; need to fix the diff vs homeo inconsistency}
+\nn{need to fix the diff vs homeo inconsistency}
-between the $n$-manifolds $X$ and $Y$ (extending a fixed diffeomorphism $\bd X \to \bd Y$).
+between the $n$-manifolds $X$ and $Y$ (extending a fixed homeomorphism $\bd X \to \bd Y$).
-For convenience, we will permit the singular cells generating $CD_*(X, Y)$ to be more general
+For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general
 than simplices --- they can be based on any linear polyhedron.
 \nn{be more restrictive here?  does more need to be said?}
-We also will use the abbreviated notation $CD_*(X) \deq CD_*(X, X)$.
+We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$.
-\begin{prop}  \label{CDprop}
+\begin{prop}  \label{CHprop}
 For $n$-manifolds $X$ and $Y$ there is a chain map
 \eq{
-e_{XY} : CD_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) .
+e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) .
 }
-On $CD_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X, Y)$ on $\bc_*(X)$
+On $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of $\Homeo(X, Y)$ on $\bc_*(X)$
 (Proposition (\ref{diff0prop})).
 For any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$,
 the following diagram commutes up to homotopy
 \eq{ \xymatrix{
-CD_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]^(.7){e_{X\sgl Y\sgl}}    & \bc_*(Y\sgl) \\
+CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]^(.7){e_{X\sgl Y\sgl}}    & \bc_*(Y\sgl) \\
-CD_*(X, Y) \otimes \bc_*(X)
+CH_*(X, Y) \otimes \bc_*(X)
 \ar@/_4ex/[r]_{e_{XY}}  \ar[u]^{\gl \otimes \gl}  &
 \bc_*(Y) \ar[u]_{\gl}
 } }
 %For any splittings $X = X_1 \cup X_2$ and $Y = Y_1 \cup Y_2$,
 %the following diagram commutes up to homotopy
 %\eq{ \xymatrix{
-%     CD_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_{XY}}    & \bc_*(Y) \\
+%     CH_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_{XY}}    & \bc_*(Y) \\
-%     CD_*(X_1, Y_1) \otimes CD_*(X_2, Y_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2)
+%     CH_*(X_1, Y_1) \otimes CH_*(X_2, Y_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2)
 %        \ar@/_4ex/[r]_{e_{X_1Y_1} \otimes e_{X_2Y_2}}  \ar[u]^{\gl \otimes \gl}  &
 %            \bc_*(Y_1) \otimes \bc_*(Y_2) \ar[u]_{\gl}
 %} }
 Any other map satisfying the above two properties is homotopic to $e_X$.
 \end{prop}
 \nn{need to rewrite for self-gluing instead of gluing two pieces together}
 \nn{Should say something stronger about uniqueness.
 Something like: there is
 a contractible subcomplex of the complex of chain maps
-$CD_*(X) \otimes \bc_*(X) \to \bc_*(X)$ (0-cells are the maps, 1-cells are homotopies, etc.),
+$CH_*(X) \otimes \bc_*(X) \to \bc_*(X)$ (0-cells are the maps, 1-cells are homotopies, etc.),
 and all choices in the construction lie in the 0-cells of this
 contractible subcomplex.
 Or maybe better to say any two choices are homotopic, and
 any two homotopies and second order homotopic, and so on.}
 Without loss of generality, we will assume $X = Y$.
 \medskip
-Let $f: P \times X \to X$ be a family of diffeomorphisms and $S \sub X$.
+Let $f: P \times X \to X$ be a family of homeomorphisms and $S \sub X$.
 We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all
-$x \notin S$ 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'
+$x \notin S$ and $p, q \in P$. Equivalently, $f$ is supported on $S$ if there is a family of homeomorphisms $f' : P \times S \to S$ and a `background'
-diffeomorphism $f_0 : X \to X$ so that
+homeomorphism $f_0 : X \to X$ so that
 \begin{align}
 	f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\
 \intertext{and}
 	f(p,x) & = f_0(x) \;\;\;\; \mbox{for}\; (p, x) \in {P \times (X \setmin S)}.
 \end{align}
 Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$.
 Let $\cU = \{U_\alpha\}$ be an open cover of $X$.
-A $k$-parameter family of diffeomorphisms $f: P \times X \to X$ is
+A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is
 {\it adapted to $\cU$} if there is a factorization
 \eq{
 P = P_1 \times \cdots \times P_m
 }
 (for some $m \le k$)
-and families of diffeomorphisms
+and families of homeomorphisms
 \eq{
 f_i :  P_i \times X \to X
 }
 such that
 \begin{itemize}
 \item each $f_i$ is supported on some connected $V_i \sub X$;
 \item the sets $V_i$ are mutually disjoint;
 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s,
 where $k_i = \dim(P_i)$; and
 \item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$
-for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$.
+for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Homeo(X)$.
 \end{itemize}
-A chain $x \in CD_k(X)$ is (by definition) adapted to $\cU$ if it is the sum
+A chain $x \in CH_k(X)$ is (by definition) adapted to $\cU$ if it is the sum
 of singular cells, each of which is adapted to $\cU$.
-(Actually, in this section we will only need families of diffeomorphisms to be
+(Actually, in this section we will only need families of homeomorphisms to be
 {\it weakly adapted} to $\cU$, meaning that the support of $f$ is contained in the union
 of at most $k$ of the $U_\alpha$'s.)
 \begin{lemma}  \label{extension_lemma}
-Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$.
+Let $x \in CH_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$.
-Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$.
+Then $x$ is homotopic (rel boundary) to some $x' \in CH_k(X)$ which is adapted to $\cU$.
 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$.
 \end{lemma}
 The proof will be given in Section \ref{sec:localising}.
 We will actually prove the following more general result.
 \end{lemma}
 \medskip
-Before diving into the details, we outline our strategy for the proof of Proposition \ref{CDprop}.
+Before diving into the details, we outline our strategy for the proof of Proposition \ref{CHprop}.
 %Suppose for the moment that evaluation maps with the advertised properties exist.
-Let $p$ be a singular cell in $CD_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$.
+Let $p$ be a singular cell in $CH_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$.
 Suppose that there exists $V \sub X$ such that
 \begin{enumerate}
 \item $V$ is homeomorphic to a disjoint union of balls, and
 \item $\supp(p) \cup \supp(b) \sub V$.
 \end{enumerate}
 Let $W = X \setmin V$, $W' = p(W)$ and $V' = X\setmin W'$.
 We then have a factorization
 \[
 	p = \gl(q, r),
 \]
-where $q \in CD_k(V, V')$ and $r \in CD_0(W, W')$.
+where $q \in CH_k(V, V')$ and $r \in CH_0(W, W')$.
 We can also factorize $b = \gl(b_V, b_W)$, where $b_V\in \bc_*(V)$ and $b_W\in\bc_0(W)$.
 According to the commutative diagram of the proposition, we must have
 \[
 	e_X(p\otimes b) = e_X(\gl(q\otimes b_V, r\otimes b_W)) =
 				gl(e_{VV'}(q\otimes b_V), e_{WW'}(r\otimes b_W)) .
 \]
-Since $r$ is a plain, 0-parameter family of diffeomorphisms, we must have
+Since $r$ is a plain, 0-parameter family of homeomorphisms, we must have
 \[
 	e_{WW'}(r\otimes b_W) = r(b_W),
 \]
-where $r(b_W)$ denotes the obvious action of diffeomorphisms on blob diagrams (in
+where $r(b_W)$ denotes the obvious action of homeomorphisms on blob diagrams (in
 this case a 0-blob diagram).
 Since $V'$ is a disjoint union of balls, $\bc_*(V')$ is acyclic in degrees $>0$
 (by \ref{disjunion} and \ref{bcontract}).
 Assuming inductively that we have already defined $e_{VV'}(\bd(q\otimes b_V))$,
 there is, up to (iterated) homotopy, a unique choice for $e_{VV'}(q\otimes b_V)$
 To show existence, we must show that the various choices involved in constructing
 evaluation maps in this way affect the final answer only by a homotopy.
 \nn{maybe put a little more into the outline before diving into the details.}
-\nn{Note: At the moment this section is very inconsistent with respect to PL versus smooth,
+\nn{Note: At the moment this section is very inconsistent with respect to PL versus smooth, etc
-homeomorphism versus diffeomorphism, etc.
 We expect that everything is true in the PL category, but at the moment our proof
 avails itself to smooth techniques.
 Furthermore, it traditional in the literature to speak of $C_*(\Diff(X))$
 rather than $C_*(\Homeo(X))$.}
 (e.g.\ $\ep_i = 2^{-i}$).
 Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$
 converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$).
 Let $\phi_l$ be an increasing sequence of positive numbers
 satisfying the inequalities of Lemma \ref{xx2phi}.
-Given a generator $p\otimes b$ of $CD_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $l$
+Given a generator $p\otimes b$ of $CH_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $l$
 define
 \[
 	N_{i,l}(p\ot b) \deq \Nbd_{l\ep_i}(|b|) \cup \Nbd_{\phi_l\delta_i}(|p|).
 \]
 In other words, we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized
 by $l$), with $\ep_i$ controlling the size of the buffers around $|b|$ and $\delta_i$ controlling
 the size of the buffers around $|p|$.
-Next we define subcomplexes $G_*^{i,m} \sub CD_*(X)\otimes \bc_*(X)$.
+Next we define subcomplexes $G_*^{i,m} \sub CH_*(X)\otimes \bc_*(X)$.
-Let $p\ot b$ be a generator of $CD_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b)
+Let $p\ot b$ be a generator of $CH_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b)
 = \deg(p) + \deg(b)$.
 $p\ot b$ is (by definition) in $G_*^{i,m}$ if either (a) $\deg(p) = 0$ or (b)
 there exist codimension-zero submanifolds $V_0,\ldots,V_m \sub X$ such that each $V_j$
 is homeomorphic to a disjoint union of balls and
 \[
 $G_*^{i,m}$ is a subcomplex where it is easy to define
 the evaluation map.
 The parameter $m$ controls the number of iterated homotopies we are able to construct
 (see Lemma \ref{m_order_hty}).
 The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of
-$CD_*(X)\ot \bc_*(X)$ (see Lemma \ref{Gim_approx}).
+$CH_*(X)\ot \bc_*(X)$ (see Lemma \ref{Gim_approx}).
 Next we define a chain map (dependent on some choices) $e: G_*^{i,m} \to \bc_*(X)$.
 Let $p\ot b \in G_*^{i,m}$.
 If $\deg(p) = 0$, define
 \[
 	e(p\ot b) = p(b) ,
 \]
-where $p(b)$ denotes the obvious action of the diffeomorphism(s) $p$ on the blob diagram $b$.
+where $p(b)$ denotes the obvious action of the homeomorphism(s) $p$ on the blob diagram $b$.
 For general $p\ot b$ ($\deg(p) \ge 1$) assume inductively that we have already defined
 $e(p'\ot b')$ when $\deg(p') + \deg(b') < k = \deg(p) + \deg(b)$.
 Choose $V = V_0$ as above so that
 \[
 	N_{i,k}(p\ot b) \subeq V \subeq N_{i,k+1}(p\ot b) .
 (The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.)
 We therefore have splittings
 \[
 	p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet b'' ,
 \]
-where $p' \in CD_*(V)$, $p'' \in CD_*(X\setmin V)$,
+where $p' \in CH_*(V)$, $p'' \in CH_*(X\setmin V)$,
 $b' \in \bc_*(V)$, $b'' \in \bc_*(X\setmin V)$,
 $f' \in \bc_*(p(V))$, and $f'' \in \bc_*(p(X\setmin V))$.
-(Note that since the family of diffeomorphisms $p$ is constant (independent of parameters)
+(Note that since the family of homeomorphisms $p$ is constant (independent of parameters)
 near $\bd V$, the expressions $p(V) \sub X$ and $p(X\setmin V) \sub X$ are
 unambiguous.)
 We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$.
 %We also have that $\deg(b'') = 0 = \deg(p'')$.
 Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$.
 There are splittings
 \[
 	p = p'_1\bullet p''_1 , \;\; b = b'_1\bullet b''_1 ,
 			\;\; e(p\ot b) - \tilde{e}(p\ot b) - h(\bd(p\ot b)) = f'_1\bullet f''_1 ,
 \]
-where $p'_1 \in CD_*(V_1)$, $p''_1 \in CD_*(X\setmin V_1)$,
+where $p'_1 \in CH_*(V_1)$, $p''_1 \in CH_*(X\setmin V_1)$,
 $b'_1 \in \bc_*(V_1)$, $b''_1 \in \bc_*(X\setmin V_1)$,
 $f'_1 \in \bc_*(p(V_1))$, and $f''_1 \in \bc_*(p(X\setmin V_1))$.
 Inductively, $\bd f'_1 = 0$ and $f_1'' = p_1''(b_1'')$.
 Choose $x'_1 \in \bc_*(p(V_1))$ so that $\bd x'_1 = f'_1$.
 Define
 Note that on $G_*^{i,m+1} \subeq G_*^{i,m}$, we have defined two maps,
 call them $e_{i,m}$ and $e_{i,m+1}$.
 An easy variation on the above lemma shows that $e_{i,m}$ and $e_{i,m+1}$ are $m$-th
 order homotopic.
-Next we show how to homotope chains in $CD_*(X)\ot \bc_*(X)$ to one of the
+Next we show how to homotope chains in $CH_*(X)\ot \bc_*(X)$ to one of the
 $G_*^{i,m}$.
 Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero.
 Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$.
-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}.
+Let $h_j: CH_*(X)\to CH_*(X)$ be a chain map homotopic to the identity whose image is spanned by homeomorphisms with support compatible with $\cU_j$, as described in Lemma \ref{xxxxx}.
 Recall that $h_j$ and also the homotopy connecting it to the identity do not increase
 supports.
 Define
 \[
 	g_j \deq h_j\circ h_{j-1} \circ \cdots \circ h_1 .
 $g_j(p)\ot b$ lies in $G_*^{i,m}$, for arbitrary $m$ and sufficiently large $i$
 (depending on $b$, $n = \deg(p)$ and $m$).
 (Note: Don't confuse this $n$ with the top dimension $n$ used elsewhere in this paper.)
 \begin{lemma} \label{Gim_approx}
-Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CD_*(X)$.
+Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CH_*(X)$.
 Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$
-there exists another constant $j_i$ such that for all $j \ge j_i$ and all $p\in CD_n(X)$
+there exists another constant $j_i$ such that for all $j \ge j_i$ and all $p\in CH_n(X)$
 we have $g_j(p)\ot b \in G_*^{i,m}$.
 \end{lemma}
 \begin{proof}
 Let $c$ be a subset of the blobs of $b$.
 \[
 	\gamma_j < \delta_i
 \]
 and also so that $\phi_t \gamma_j$ is less than the constant $\rho(M)$ of Lemma \ref{xxzz11}.
-Let $j \ge j_i$ and $p\in CD_n(X)$.
+Let $j \ge j_i$ and $p\in CH_n(X)$.
 Let $q$ be a generator appearing in $g_j(p)$.
 Note that $|q|$ is contained in a union of $n$ elements of the cover $\cU_j$,
 which implies that $|q|$ is contained in a union of $n$ metric balls of radius $\delta_i$.
 We must show that $q\ot b \in G_*^{i,m}$, which means finding neighborhoods
 $V_0,\ldots,V_m \sub X$ of $|q|\cup |b|$ such that each $V_j$
 \nn{outline of what remains to be done:}
 \begin{itemize}
 \item We need to assemble the maps for the various $G^{i,m}$ into
-a map for all of $CD_*\ot \bc_*$.
+a map for all of $CH_*\ot \bc_*$.
-One idea: Think of the $g_j$ as a sort of homotopy (from $CD_*\ot \bc_*$ to itself)
+One idea: Think of the $g_j$ as a sort of homotopy (from $CH_*\ot \bc_*$ to itself)
-parameterized by $[0,\infty)$.  For each $p\ot b$ in $CD_*\ot \bc_*$ choose a sufficiently
+parameterized by $[0,\infty)$.  For each $p\ot b$ in $CH_*\ot \bc_*$ choose a sufficiently
 large $j'$.  Use these choices to reparameterize $g_\bullet$ so that each
 $p\ot b$ gets pushed as far as the corresponding $j'$.
 \item Independence of metric, $\ep_i$, $\delta_i$:
 For a different metric etc. let $\hat{G}^{i,m}$ denote the alternate subcomplexes
 and $\hat{N}_{i,l}$ the alternate neighborhoods.

changeset 236	3feb6e24a518
parent 213	a60332c29d0b
child 244	cf01e213044a