blob: comparison text/evmap.tex

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-:63ddcff52748
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 \nn{should comment at the start about any assumptions about smooth, PL etc.}
 Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of
 the space of homeomorphisms
-between the $n$-manifolds $X$ and $Y$ (extending a fixed homeomorphism $\bd X \to \bd Y$).
+between the $n$-manifolds $X$ and $Y$ (any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$).
 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$.
 (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?})
 \begin{enumerate}
 \item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of
 $\Homeo(X, Y)$ on $\bc_*(X)$ (Property (\ref{property:functoriality})), and
 \item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$,
 the following diagram commutes up to homotopy
-\eq{ \xymatrix{
+\begin{equation*}
-CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]^(.7){e_{X\sgl Y\sgl}}  \ar[d]^{\gl \otimes \gl}   & \bc_*(Y\sgl)  \ar[d]_{\gl} \\
+\xymatrix@C+2cm{
 CH_*(X, Y) \otimes \bc_*(X)
-\ar@/_4ex/[r]_{e_{XY}}   &
+\ar[r]_(.6){e_{XY}}  \ar[d]^{\gl \otimes \gl}   &
-\bc_*(Y)
+\bc_*(Y)\ar[d]^{\gl} \\
-} }
+CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]_(.6){e_{X\sgl Y\sgl}}   & 	\bc_*(Y\sgl)
+}
+\end{equation*}
 \end{enumerate}
 Up to (iterated) homotopy, there is a unique family $\{e_{XY}\}$ of chain maps
 satisfying the above two conditions.
 \end{prop}
 The proof will be given in \S\ref{sec:localising}.
 \medskip
 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 $CH_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$.
 We say that $p\ot b$ is {\it localizable} if there exists $V \sub X$ such that
 \begin{itemize}
 \item $V$ is homeomorphic to a disjoint union of balls, and
 \item $\supp(p) \cup \supp(b) \sub V$.
 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 homeomorphisms, we must have
+Since $r$ is a  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 homeomorphisms on blob diagrams (in
 this case a 0-blob diagram).
 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.
 Now for a little more detail.
 (But we're still just motivating the full, gory details, which will follow.)
-Choose a metric on $X$, and let $\cU_\gamma$ be the open cover of by balls of radius $\gamma$.
+Choose a metric on $X$, and let $\cU_\gamma$ be the open cover of balls of radius $\gamma$.
 By Lemma \ref{extension_lemma} we can restrict our attention to $k$-parameter families
 $p$ of homeomorphisms such that $\supp(p)$ is contained in the union of $k$ $\gamma$-balls.
 For fixed blob diagram $b$ and fixed $k$, it's not hard to show that for $\gamma$ small enough
 $p\ot b$ must be localizable.
 On the other hand, for fixed $k$ and $\gamma$ there exist $p$ and $b$ such that $p\ot b$ is not localizable,
 The construction of $e_X$, as outlined above, depends on various choices, one of which
 is the choice, for each localizable generator $p\ot b$,
 of disjoint balls $V$ containing $\supp(p)\cup\supp(b)$.
 Let $V'$ be another disjoint union of balls containing $\supp(p)\cup\supp(b)$,
-and assume that there exists yet another disjoint union of balls $W$ with $W$ containing
+and assume that there exists yet another disjoint union of balls $W$ containing
 $V\cup V'$.
 Then we can use $W$ to construct a homotopy between the two versions of $e_X$
 associated to $V$ and $V'$.
 If we impose no constraints on $V$ and $V'$ then such a $W$ need not exist.
 Thus we will insist below that $V$ (and $V'$) be contained in small metric neighborhoods
 \medskip
 \begin{proof}[Proof of Proposition \ref{CHprop}.]
-Notation: Let $|b| = \supp(b)$, $|p| = \supp(p)$.
+We'll use the notation $|b| = \supp(b)$ and $|p| = \supp(p)$.
 Choose a metric on $X$.
-Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero
+Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging monotonically to zero
 (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}.
+satisfying the inequalities of Lemma \ref{xx2phi} below.
 Given a generator $p\otimes b$ of $CH_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $l$
 define
 \[
 	N_{i,l}(p\ot b) \deq \Nbd_{l\ep_i}(|b|) \cup \Nbd_{\phi_l\delta_i}(|p|).
 \]
 the size of the buffers around $|p|$.
 Next we define subcomplexes $G_*^{i,m} \sub CH_*(X)\otimes \bc_*(X)$.
 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)
+We say $p\ot b$ is in $G_*^{i,m}$ exactly when 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
 \[
 	N_{i,k}(p\ot b) \subeq V_0 \subeq N_{i,k+1}(p\ot b)
 			\subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,k+m+1}(p\ot b) .
 \]
-Further, we require (inductively) that $\bd(p\ot b) \in G_*^{i,m}$.
+and further $\bd(p\ot b) \in G_*^{i,m}$.
 We also require that $b$ is splitable (transverse) along the boundary of each $V_l$.
 Note that $G_*^{i,m+1} \subeq G_*^{i,m}$.
 As sketched above and explained in detail below,
 \begin{lemma} \label{m_order_hty}
 Let $\tilde{e} :  G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with
 different choices of $V$ (and hence also different choices of $x'$) at each step.
 If $m \ge 1$ then $e$ and $\tilde{e}$ are homotopic.
 If $m \ge 2$ then any two choices of this first-order homotopy are second-order homotopic.
-And so on.
+Continuing, $e :  G_*^{i,m} \to \bc_*(X)$ is well-defined up to $m$-th order homotopy.
-In other words,  $e :  G_*^{i,m} \to \bc_*(X)$ is well-defined up to $m$-th order homotopy.
 \end{lemma}
 \begin{proof}
 We construct $h: G_*^{i,m} \to \bc_*(X)$ such that $\bd h + h\bd = e - \tilde{e}$.
-$e$ and $\tilde{e}$ coincide on bidegrees $(0, j)$, so define $h$
+The chain maps $e$ and $\tilde{e}$ coincide on bidegrees $(0, j)$, so define $h$
 to be zero there.
 Assume inductively that $h$ has been defined for degrees less than $k$.
 Let $p\ot b$ be a generator of degree $k$.
 Choose $V_1$ as in the definition of $G_*^{i,m}$ so that
 \[
 \]
 for all $l\ge m$ and all $q\ot a$ which appear in the boundary of $p\ot b$.
 \begin{proof}
-Let $c$ be a subset of the blobs of $b$.
-There exists $\lambda > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$
+There exists $\lambda > 0$ such that for every  subset $c$ of the blobs of $b$ $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ .
-and all such $c$.
+(Here we are using the fact that the blobs are piecewise-linear and thatthat $\bd c$ is collared.)
-(Here we are using a piecewise smoothness assumption for $\bd c$, and also
-the fact that $\bd c$ is collared.
 We need to consider all such $c$ because all generators appearing in
 iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.)
 Let $r = \deg(b)$ and
 \[
 The same argument shows that each generator involved in iterated boundaries of $q\ot b$
 is in $G_*^{i,m}$.
 \end{proof}
-In the next few lemmas we have made no effort to optimize the various bounds.
+In the next three lemmas, which provide the estimates needed above, we have made no effort to optimize the various bounds.
 (The bounds are, however, optimal in the sense of minimizing the amount of work
 we do.  Equivalently, they are the first bounds we thought of.)
 We say that a subset $S$ of a metric space has radius $\le r$ if $S$ is contained in
 some metric ball of radius $r$.
 \begin{proof} \label{xxyy2}
 Let $S$ be contained in $B_r(y)$, $y \in \ebb^n$.
 Note that if $a \ge 2r$ then $\Nbd_a(S) \sup B_r(y)$.
 Let $z\in \Nbd_a(S) \setmin B_r(y)$.
 Consider the triangle
-\nn{give figure?} with vertices $z$, $y$ and $s$ with $s\in S$.
+with vertices $z$, $y$ and $s$ with $s\in S$.
 The length of the edge $yz$ is greater than $r$ which is greater
 than the length of the edge $ys$.
 It follows that the angle at $z$ is less than $\pi/2$ (less than $\pi/3$, in fact),
 which means that points on the edge $yz$ near $z$ are closer to $s$ than $z$ is,
 which implies that these points are also in $\Nbd_a(S)$.
 Hence $\Nbd_a(S)$ is star-shaped with respect to $y$.
 \end{proof}
 If we replace $\ebb^n$ above with an arbitrary compact Riemannian manifold $M$,
 the same result holds, so long as $a$ is not too large:
-\nn{what about PL? TOP?}
+\nn{replace this with a PL version}
 \begin{lemma} \label{xxzz11}
 Let $M$ be a compact Riemannian manifold.
 Then there is a constant $\rho(M)$ such that for all
 subsets $S\sub M$ of radius $\le r$ and all $a$ such that $2r \le a \le \rho(M)$,
 	U \subeq \Nbd_{\phi_{k-1}r'}(R) = \Nbd_{t}(S) \subeq \Nbd_{\phi_k r}(S) ,
 \]
 where $t = \phi_{k-1}(2\phi_{k-1}+2)r + \phi_{k-1} r$.
 \end{proof}
-\medskip
+We now return to defining the chain maps $e_X$.
 Let $R_*$ be the chain complex with a generating 0-chain for each non-negative
 integer and a generating 1-chain connecting each adjacent pair $(j, j+1)$.
 (So $R_*$ is a simplicial version of the non-negative reals.)
 Denote the 0-chains by $j$ (for $j$ a non-negative integer) and the 1-chain connecting $j$ and $j+1$
 Next we show that the action maps are compatible with gluing.
 Let $G^m_*$ and $\ol{G}^m_*$ be the complexes, as above, used for defining
 the action maps $e_{X\sgl}$ and $e_X$.
 The gluing map $X\sgl\to X$ induces a map
 \[
-	\gl: R_*\ot CH_*(X\sgl, X \sgl) \otimes \bc_*(X \sgl) \to R_*\ot CH_*(X, X) \otimes \bc_*(X) ,
+	\gl:  R_*\ot CH_*(X, X) \otimes \bc_*(X)  \to R_*\ot CH_*(X\sgl, X \sgl) \otimes \bc_*(X \sgl) ,
 \]
 and it is easy to see that $\gl(G^m_*)\sub \ol{G}^m_*$.
 From this it follows that the diagram in the statement of Proposition \ref{CHprop} commutes.
-\medskip
+\todo{this paragraph isn't very convincing, or at least I don't see what's going on}
 Finally we show that the action maps defined above are independent of
 the choice of metric (up to iterated homotopy).
 The arguments are very similar to ones given above, so we only sketch them.
 Let $g$ and $g'$ be two metrics on $X$, and let $e$ and $e'$ be the corresponding
 actions $CH_*(X, X) \ot \bc_*(X)\to\bc_*(X)$.
 by $p\ot b$ such that $|p|\cup|b|$ is contained in a disjoint union of balls.
 Using acyclic models, we can construct a homotopy from $e$ to $e'$ on $F_*$.
 We now observe that $CH_*(X, X) \ot \bc_*(X)$ retracts to $F_*$.
 Similar arguments show that this homotopy from $e$ to $e'$ is well-defined
 up to second order homotopy, and so on.
+This completes the proof of Proposition \ref{CHprop}.
 \end{proof}
 \begin{rem*}
 \label{rem:for-small-blobs}
 For the proof of Lemma \ref{lem:CH-small-blobs} below we will need the following observation on the action constructed above.
 Let $b$ be a blob diagram and $p:P\times X\to X$ be a family of homeomorphisms.
 Then we may choose $e$ such that $e(p\ot b)$ is a sum of generators, each
 of which has support close to $p(t,|b|)$ for some $t\in P$.
-More precisely, the support of the generators is contained in a small neighborhood
+More precisely, the support of the generators is contained in the union of a small neighborhood
-of $p(t,|b|)$ union some small balls.
+of $p(t,|b|)$ with some small balls.
 (Here ``small" is in terms of the metric on $X$ that we chose to construct $e$.)
 \end{rem*}
 \begin{prop}

changeset 430	c5a35886cd82
parent 426	8aca80203f9d
child 434	785e4953a811
child 437	93ce0ba3d2d7