author | Scott Morrison <scott@tqft.net> |
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\section{Action of \texorpdfstring{$\CH{X}$}{$C_*(Homeo(M))$}} |
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\label{sec:evaluation} |
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\nn{should comment at the start about any assumptions about smooth, PL etc.} |
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\nn{should maybe mention alternate def of blob complex (sort-of-simplicial space instead of |
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sort-of-simplicial set) where this action would be easy} |
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Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of |
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the space of homeomorphisms |
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between the $n$-manifolds $X$ and $Y$ (any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). |
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We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
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(For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
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than simplices --- they can be based on any linear polyhedron. |
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\nn{be more restrictive here? does more need to be said?}) |
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\begin{thm} \label{thm:CH} |
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For $n$-manifolds $X$ and $Y$ there is a chain map |
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\eq{ |
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e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) |
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} |
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such that |
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\begin{enumerate} |
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\item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of |
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$\Homeo(X, Y)$ on $\bc_*(X)$ described in Property (\ref{property:functoriality}), and |
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\item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
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the following diagram commutes up to homotopy |
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\begin{equation*} |
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\xymatrix@C+2cm{ |
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CH_*(X, Y) \otimes \bc_*(X) |
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\ar[r]_(.6){e_{XY}} \ar[d]^{\gl \otimes \gl} & |
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\bc_*(Y)\ar[d]^{\gl} \\ |
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CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]_(.6){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) |
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} |
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\end{equation*} |
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\end{enumerate} |
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Moreover, for any $m \geq 0$, we can find a family of chain maps $\{e_{XY}\}$ |
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satisfying the above two conditions which is $m$-connected. In particular, this means that the choice of chain map above is unique up to homotopy. |
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\end{thm} |
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\begin{rem} |
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Note that the statement doesn't quite give uniqueness up to iterated homotopy. We fully expect that this should actually be the case, but haven't been able to prove this. |
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\end{rem} |
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Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma}, |
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and then give an outline of the method of proof. |
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Without loss of generality, we will assume $X = Y$. |
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\medskip |
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Let $f: P \times X \to X$ be a family of homeomorphisms (e.g. a generator of $CH_*(X)$) |
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and let $S \sub X$. |
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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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$x \notin S$ and $p, q \in P$. Equivalently, $f$ is supported on $S$ if |
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there is a family of homeomorphisms $f' : P \times S \to S$ and a ``background" |
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homeomorphism $f_0 : X \to X$ so that |
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\begin{align*} |
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f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\ |
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\intertext{and} |
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f(p,x) & = f_0(x) \;\;\;\; \mbox{for}\; (p, x) \in {P \times (X \setmin S)}. |
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\end{align*} |
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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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(So when we talk about ``the" support of a family, there is some ambiguity, |
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but this ambiguity will not matter to us.) |
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Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
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A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is |
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{\it adapted to $\cU$} |
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if the support of $f$ is contained in the union |
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of at most $k$ of the $U_\alpha$'s. |
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\begin{lemma} \label{extension_lemma} |
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Let $x \in CH_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
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Then $x$ is homotopic (rel boundary) to some $x' \in CH_k(X)$ which is adapted to $\cU$. |
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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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\end{lemma} |
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The proof will be given in \S\ref{sec:localising}. |
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\medskip |
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Before diving into the details, we outline our strategy for the proof of Theorem \ref{thm:CH}. |
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Let $p$ be a singular cell in $CH_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
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We say that $p\ot b$ is {\it localizable} if there exists $V \sub X$ such that |
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\begin{itemize} |
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\item $V$ is homeomorphic to a disjoint union of balls, and |
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\item $\supp(p) \cup \supp(b) \sub V$. |
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\end{itemize} |
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(Recall that $\supp(b)$ is defined to be the union of the blobs of the diagram $b$.) |
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Assuming that $p\ot b$ is localizable as above, |
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let $W = X \setmin V$, $W' = p(W)$ and $V' = X\setmin W'$. |
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We then have a factorization |
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\[ |
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p = \gl(q, r), |
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\] |
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where $q \in CH_k(V, V')$ and $r \in CH_0(W, W')$. |
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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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According to the commutative diagram of the proposition, we must have |
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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 0-parameter family of homeomorphisms, we must have |
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\[ |
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e_{WW'}(r\otimes b_W) = r(b_W), |
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\] |
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where $r(b_W)$ denotes the obvious action of homeomorphisms 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 Properties \ref{property:disjoint-union} and \ref{property:contractibility}). |
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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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\[ |
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\bd(e_{VV'}(q\otimes b_V)) = e_{VV'}(\bd(q\otimes b_V)) . |
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\] |
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Thus the conditions of the proposition determine (up to homotopy) the evaluation |
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map for localizable generators $p\otimes b$. |
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On the other hand, Lemma \ref{extension_lemma} allows us to homotope |
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arbitrary generators to sums of localizable generators. |
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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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Now for a little more detail. |
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(But we're still just motivating the full, gory details, which will follow.) |
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Choose a metric on $X$, and let $\cU_\gamma$ be the open cover of $X$ by balls of radius $\gamma$. |
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By Lemma \ref{extension_lemma} we can restrict our attention to $k$-parameter families |
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$p$ of homeomorphisms such that $\supp(p)$ is contained in the union of $k$ $\gamma$-balls. |
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For fixed blob diagram $b$ and fixed $k$, it's not hard to show that for $\gamma$ small enough |
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$p\ot b$ must be localizable. |
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On the other hand, for fixed $k$ and $\gamma$ there exist $p$ and $b$ such that $p\ot b$ is not localizable, |
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and for fixed $\gamma$ and $b$ there exist non-localizable $p\ot b$ for sufficiently large $k$. |
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Thus we will need to take an appropriate limit as $\gamma$ approaches zero. |
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The construction of $e_X$, as outlined above, depends on various choices, one of which |
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is the choice, for each localizable generator $p\ot b$, |
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of disjoint balls $V$ containing $\supp(p)\cup\supp(b)$. |
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Let $V'$ be another disjoint union of balls containing $\supp(p)\cup\supp(b)$, |
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and assume that there exists yet another disjoint union of balls $W$ containing |
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$V\cup V'$. |
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Then we can use $W$ to construct a homotopy between the two versions of $e_X$ |
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associated to $V$ and $V'$. |
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If we impose no constraints on $V$ and $V'$ then such a $W$ need not exist. |
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Thus we will insist below that $V$ (and $V'$) be contained in small metric neighborhoods |
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of $\supp(p)\cup\supp(b)$. |
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Because we want not mere homotopy uniqueness but iterated homotopy uniqueness, |
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we will similarly require that $W$ be contained in a slightly larger metric neighborhood of |
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$\supp(p)\cup\supp(b)$, and so on. |
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\begin{proof}[Proof of Theorem \ref{thm:CH}.] |
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We'll use the notation $|b| = \supp(b)$ and $|p| = \supp(p)$. |
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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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Let $\phi_l$ be an increasing sequence of positive numbers |
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satisfying the inequalities of Lemma \ref{xx2phi} below. |
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Given a generator $p\otimes b$ of $CH_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $l$ |
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define |
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\[ |
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N_{i,l}(p\ot b) \deq \Nbd_{l\ep_i}(|b|) \cup \Nbd_{\phi_l\delta_i}(|p|). |
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\] |
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In other words, for each $i$ |
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we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized |
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by $l$), with $\ep_i$ controlling the size of the buffers around $|b|$ and $\delta_i$ controlling |
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the size of the buffers around $|p|$. |
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Next we define subcomplexes $G_*^{i,m} \sub CH_*(X)\otimes \bc_*(X)$. |
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Let $p\ot b$ be a generator of $CH_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b) |
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= \deg(p) + \deg(b)$. |
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We say $p\ot b$ is in $G_*^{i,m}$ exactly when either (a) $\deg(p) = 0$ or (b) |
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there exist codimension-zero submanifolds $V_0,\ldots,V_m \sub X$ such that each $V_j$ |
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is homeomorphic to a disjoint union of balls and |
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\[ |
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N_{i,k}(p\ot b) \subeq V_0 \subeq N_{i,k+1}(p\ot b) |
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\subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,k+m+1}(p\ot b) , |
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\] |
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and further $\bd(p\ot b) \in G_*^{i,m}$. |
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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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(see Lemma \ref{m_order_hty}). |
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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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$CH_*(X)\ot \bc_*(X)$ (see Lemma \ref{Gim_approx}). |
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Next we define a chain map (dependent on some choices) $e_{i,m}: G_*^{i,m} \to \bc_*(X)$. |
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(When the domain is clear from context we will drop the subscripts and write |
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simply $e: G_*^{i,m} \to \bc_*(X)$). |
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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 homeomorphism(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 $V = V_0$ as above so that |
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\[ |
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N_{i,k}(p\ot b) \subeq V \subeq N_{i,k+1}(p\ot b) . |
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\] |
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Let $\bd(p\ot b) = \sum_j p_j\ot b_j$, and let $V^j$ 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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\[ |
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V^j \subeq N_{i,k}(p_j\ot b_j) \subeq N_{i,k}(p\ot b) \subeq V . |
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\] |
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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 $p' \in CH_*(V)$, $p'' \in CH_*(X\setmin V)$, |
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$b' \in \bc_*(V)$, $b'' \in \bc_*(X\setmin V)$, |
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$f' \in \bc_*(p(V))$, and $f'' \in \bc_*(p(X\setmin V))$. |
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(Note that since the family of homeomorphisms $p$ is constant (independent of parameters) |
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near $\bd V$, the expressions $p(V) \sub X$ and $p(X\setmin V) \sub X$ are |
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unambiguous.) |
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We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$. |
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%We also have that $\deg(b'') = 0 = \deg(p'')$. |
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Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$. |
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This is possible by Properties \ref{property:disjoint-union} and \ref{property:contractibility} and the fact that isotopic fields |
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differ by a local relation. |
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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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Note that above we are essentially using the method of acyclic models \nn{\S \ref{sec:moam}}. |
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For each generator $p\ot b$ we specify the acyclic (in positive degrees) |
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target complex $\bc_*(p(V)) \bullet p''(b'')$. |
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The definition of $e: G_*^{i,m} \to \bc_*(X)$ depends on two sets of choices: |
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The choice of neighborhoods $V$ and the choice of inverse boundaries $x'$. |
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The next lemma shows that up to (iterated) homotopy $e$ is independent |
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of these choices. |
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(Note that independence of choices of $x'$ (for fixed choices of $V$) |
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is a standard result in the method of acyclic models.) |
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%\begin{lemma} |
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%Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with |
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%different choices of $x'$ at each step. |
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%(Same choice of $V$ at each step.) |
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%Then $e$ and $\tilde{e}$ are homotopic via a homotopy in $\bc_*(p(V)) \bullet p''(b'')$. |
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%Any two choices of such a first-order homotopy are second-order homotopic, and so on, |
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%to arbitrary order. |
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%\end{lemma} |
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%\begin{proof} |
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%This is a standard result in the method of acyclic models. |
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%\nn{should we say more here?} |
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%\nn{maybe this lemma should be subsumed into the next lemma. probably it should.} |
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%\end{proof} |
|
84 | 267 |
|
87 | 268 |
\begin{lemma} \label{m_order_hty} |
84 | 269 |
Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with |
270 |
different choices of $V$ (and hence also different choices of $x'$) at each step. |
|
271 |
If $m \ge 1$ then $e$ and $\tilde{e}$ are homotopic. |
|
272 |
If $m \ge 2$ then any two choices of this first-order homotopy are second-order homotopic. |
|
430 | 273 |
Continuing, $e : G_*^{i,m} \to \bc_*(X)$ is well-defined up to $m$-th order homotopy. |
84 | 274 |
\end{lemma} |
275 |
||
276 |
\begin{proof} |
|
277 |
We construct $h: G_*^{i,m} \to \bc_*(X)$ such that $\bd h + h\bd = e - \tilde{e}$. |
|
430 | 278 |
The chain maps $e$ and $\tilde{e}$ coincide on bidegrees $(0, j)$, so define $h$ |
84 | 279 |
to be zero there. |
280 |
Assume inductively that $h$ has been defined for degrees less than $k$. |
|
281 |
Let $p\ot b$ be a generator of degree $k$. |
|
282 |
Choose $V_1$ as in the definition of $G_*^{i,m}$ so that |
|
283 |
\[ |
|
284 |
N_{i,k+1}(p\ot b) \subeq V_1 \subeq N_{i,k+2}(p\ot b) . |
|
285 |
\] |
|
286 |
There are splittings |
|
287 |
\[ |
|
288 |
p = p'_1\bullet p''_1 , \;\; b = b'_1\bullet b''_1 , |
|
289 |
\;\; e(p\ot b) - \tilde{e}(p\ot b) - h(\bd(p\ot b)) = f'_1\bullet f''_1 , |
|
290 |
\] |
|
236 | 291 |
where $p'_1 \in CH_*(V_1)$, $p''_1 \in CH_*(X\setmin V_1)$, |
84 | 292 |
$b'_1 \in \bc_*(V_1)$, $b''_1 \in \bc_*(X\setmin V_1)$, |
293 |
$f'_1 \in \bc_*(p(V_1))$, and $f''_1 \in \bc_*(p(X\setmin V_1))$. |
|
88 | 294 |
Inductively, $\bd f'_1 = 0$ and $f_1'' = p_1''(b_1'')$. |
84 | 295 |
Choose $x'_1 \in \bc_*(p(V_1))$ so that $\bd x'_1 = f'_1$. |
296 |
Define |
|
297 |
\[ |
|
298 |
h(p\ot b) \deq x'_1 \bullet p''_1(b''_1) . |
|
299 |
\] |
|
300 |
This completes the construction of the first-order homotopy when $m \ge 1$. |
|
301 |
||
302 |
The $j$-th order homotopy is constructed similarly, with $V_j$ replacing $V_1$ above. |
|
303 |
\end{proof} |
|
304 |
||
305 |
Note that on $G_*^{i,m+1} \subeq G_*^{i,m}$, we have defined two maps, |
|
249 | 306 |
$e_{i,m}$ and $e_{i,m+1}$. |
307 |
An easy variation on the above lemma shows that |
|
308 |
the restrictions of $e_{i,m}$ and $e_{i,m+1}$ to $G_*^{i,m+1}$ are $m$-th |
|
84 | 309 |
order homotopic. |
310 |
||
236 | 311 |
Next we show how to homotope chains in $CH_*(X)\ot \bc_*(X)$ to one of the |
85 | 312 |
$G_*^{i,m}$. |
313 |
Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero. |
|
314 |
Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$. |
|
345 | 315 |
Let $h_j: CH_*(X)\to CH_*(X)$ be a chain map homotopic to the identity whose image is |
316 |
spanned by families of homeomorphisms with support compatible with $\cU_j$, |
|
317 |
as described in Lemma \ref{extension_lemma}. |
|
86 | 318 |
Recall that $h_j$ and also the homotopy connecting it to the identity do not increase |
85 | 319 |
supports. |
320 |
Define |
|
321 |
\[ |
|
322 |
g_j \deq h_j\circ h_{j-1} \circ \cdots \circ h_1 . |
|
323 |
\] |
|
324 |
The next lemma says that for all generators $p\ot b$ we can choose $j$ large enough so that |
|
325 |
$g_j(p)\ot b$ lies in $G_*^{i,m}$, for arbitrary $m$ and sufficiently large $i$ |
|
247 | 326 |
(depending on $b$, $\deg(p)$ and $m$). |
327 |
%(Note: Don't confuse this $n$ with the top dimension $n$ used elsewhere in this paper.) |
|
85 | 328 |
|
87 | 329 |
\begin{lemma} \label{Gim_approx} |
236 | 330 |
Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CH_*(X)$. |
85 | 331 |
Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$ |
255 | 332 |
there exists another constant $j_{ibmn}$ such that for all $j \ge j_{ibmn}$ and all $p\in CH_n(X)$ |
85 | 333 |
we have $g_j(p)\ot b \in G_*^{i,m}$. |
334 |
\end{lemma} |
|
335 |
||
255 | 336 |
For convenience we also define $k_{bmp} = k_{bmn}$ |
337 |
and $j_{ibmp} = j_{ibmn}$ where $n=\deg(p)$. |
|
254 | 338 |
Note that we may assume that |
339 |
\[ |
|
340 |
k_{bmp} \ge k_{alq} |
|
341 |
\] |
|
342 |
for all $l\ge m$ and all $q\ot a$ which appear in the boundary of $p\ot b$. |
|
255 | 343 |
Additionally, we may assume that |
344 |
\[ |
|
345 |
j_{ibmp} \ge j_{ialq} |
|
346 |
\] |
|
347 |
for all $l\ge m$ and all $q\ot a$ which appear in the boundary of $p\ot b$. |
|
348 |
||
254 | 349 |
|
85 | 350 |
\begin{proof} |
430 | 351 |
|
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|
352 |
There exists $\lambda > 0$ such that for every subset $c$ of the blobs of $b$ the set $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ . |
434 | 353 |
(Here we are using the fact that the blobs are |
354 |
piecewise smooth or piecewise-linear and that $\bd c$ is collared.) |
|
90 | 355 |
We need to consider all such $c$ because all generators appearing in |
247 | 356 |
iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.) |
85 | 357 |
|
86 | 358 |
Let $r = \deg(b)$ and |
359 |
\[ |
|
90 | 360 |
t = r+n+m+1 = \deg(p\ot b) + m + 1. |
86 | 361 |
\] |
85 | 362 |
|
363 |
Choose $k = k_{bmn}$ such that |
|
364 |
\[ |
|
248 | 365 |
t\ep_k < \lambda |
85 | 366 |
\] |
367 |
and |
|
368 |
\[ |
|
90 | 369 |
n\cdot (2 (\phi_t + 1) \delta_k) < \ep_k . |
85 | 370 |
\] |
371 |
Let $i \ge k_{bmn}$. |
|
372 |
Choose $j = j_i$ so that |
|
373 |
\[ |
|
90 | 374 |
\gamma_j < \delta_i |
375 |
\] |
|
376 |
and also so that $\phi_t \gamma_j$ is less than the constant $\rho(M)$ of Lemma \ref{xxzz11}. |
|
377 |
||
236 | 378 |
Let $j \ge j_i$ and $p\in CH_n(X)$. |
90 | 379 |
Let $q$ be a generator appearing in $g_j(p)$. |
380 |
Note that $|q|$ is contained in a union of $n$ elements of the cover $\cU_j$, |
|
381 |
which implies that $|q|$ is contained in a union of $n$ metric balls of radius $\delta_i$. |
|
382 |
We must show that $q\ot b \in G_*^{i,m}$, which means finding neighborhoods |
|
383 |
$V_0,\ldots,V_m \sub X$ of $|q|\cup |b|$ such that each $V_j$ |
|
384 |
is homeomorphic to a disjoint union of balls and |
|
385 |
\[ |
|
386 |
N_{i,n}(q\ot b) \subeq V_0 \subeq N_{i,n+1}(q\ot b) |
|
387 |
\subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,t}(q\ot b) . |
|
388 |
\] |
|
248 | 389 |
Recall that |
390 |
\[ |
|
391 |
N_{i,a}(q\ot b) \deq \Nbd_{a\ep_i}(|b|) \cup \Nbd_{\phi_a\delta_i}(|q|). |
|
392 |
\] |
|
90 | 393 |
By repeated applications of Lemma \ref{xx2phi} we can find neighborhoods $U_0,\ldots,U_m$ |
394 |
of $|q|$, each homeomorphic to a disjoint union of balls, with |
|
395 |
\[ |
|
396 |
\Nbd_{\phi_{n+l} \delta_i}(|q|) \subeq U_l \subeq \Nbd_{\phi_{n+l+1} \delta_i}(|q|) . |
|
85 | 397 |
\] |
248 | 398 |
The inequalities above guarantee that |
399 |
for each $0\le l\le m$ we can find $u_l$ with |
|
90 | 400 |
\[ |
401 |
(n+l)\ep_i \le u_l \le (n+l+1)\ep_i |
|
402 |
\] |
|
403 |
such that each component of $U_l$ is either disjoint from $\Nbd_{u_l}(|b|)$ or contained in |
|
404 |
$\Nbd_{u_l}(|b|)$. |
|
405 |
This is because there are at most $n$ components of $U_l$, and each component |
|
406 |
has radius $\le (\phi_t + 1) \delta_i$. |
|
407 |
It follows that |
|
408 |
\[ |
|
409 |
V_l \deq \Nbd_{u_l}(|b|) \cup U_l |
|
410 |
\] |
|
411 |
is homeomorphic to a disjoint union of balls and satisfies |
|
412 |
\[ |
|
413 |
N_{i,n+l}(q\ot b) \subeq V_l \subeq N_{i,n+l+1}(q\ot b) . |
|
414 |
\] |
|
86 | 415 |
|
90 | 416 |
The same argument shows that each generator involved in iterated boundaries of $q\ot b$ |
417 |
is in $G_*^{i,m}$. |
|
86 | 418 |
\end{proof} |
419 |
||
430 | 420 |
In the next three lemmas, which provide the estimates needed above, we have made no effort to optimize the various bounds. |
86 | 421 |
(The bounds are, however, optimal in the sense of minimizing the amount of work |
422 |
we do. Equivalently, they are the first bounds we thought of.) |
|
423 |
||
424 |
We say that a subset $S$ of a metric space has radius $\le r$ if $S$ is contained in |
|
425 |
some metric ball of radius $r$. |
|
426 |
||
427 |
\begin{lemma} |
|
428 |
Let $S \sub \ebb^n$ (Euclidean $n$-space) have radius $\le r$. |
|
429 |
Then $\Nbd_a(S)$ is homeomorphic to a ball for $a \ge 2r$. |
|
430 |
\end{lemma} |
|
431 |
||
432 |
\begin{proof} \label{xxyy2} |
|
433 |
Let $S$ be contained in $B_r(y)$, $y \in \ebb^n$. |
|
89 | 434 |
Note that if $a \ge 2r$ then $\Nbd_a(S) \sup B_r(y)$. |
435 |
Let $z\in \Nbd_a(S) \setmin B_r(y)$. |
|
436 |
Consider the triangle |
|
494
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Scott Morrison <scott@tqft.net>
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492
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|
437 |
with vertices $z$, $y$ and $s$ with $s\in S$ such that $z \in B_a(s)$. |
89 | 438 |
The length of the edge $yz$ is greater than $r$ which is greater |
439 |
than the length of the edge $ys$. |
|
440 |
It follows that the angle at $z$ is less than $\pi/2$ (less than $\pi/3$, in fact), |
|
441 |
which means that points on the edge $yz$ near $z$ are closer to $s$ than $z$ is, |
|
442 |
which implies that these points are also in $\Nbd_a(S)$. |
|
443 |
Hence $\Nbd_a(S)$ is star-shaped with respect to $y$. |
|
444 |
\end{proof} |
|
445 |
||
446 |
If we replace $\ebb^n$ above with an arbitrary compact Riemannian manifold $M$, |
|
447 |
the same result holds, so long as $a$ is not too large: |
|
430 | 448 |
\nn{replace this with a PL version} |
89 | 449 |
|
450 |
\begin{lemma} \label{xxzz11} |
|
451 |
Let $M$ be a compact Riemannian manifold. |
|
452 |
Then there is a constant $\rho(M)$ such that for all |
|
453 |
subsets $S\sub M$ of radius $\le r$ and all $a$ such that $2r \le a \le \rho(M)$, |
|
454 |
$\Nbd_a(S)$ is homeomorphic to a ball. |
|
455 |
\end{lemma} |
|
456 |
||
457 |
\begin{proof} |
|
458 |
Choose $\rho = \rho(M)$ such that $3\rho/2$ is less than the radius of injectivity of $M$, |
|
459 |
and also so that for any point $y\in M$ the geodesic coordinates of radius $3\rho/2$ around |
|
460 |
$y$ distort angles by only a small amount. |
|
461 |
Now the argument of the previous lemma works. |
|
85 | 462 |
\end{proof} |
463 |
||
464 |
||
89 | 465 |
|
466 |
\begin{lemma} \label{xx2phi} |
|
467 |
Let $S \sub M$ be contained in a union (not necessarily disjoint) |
|
86 | 468 |
of $k$ metric balls of radius $r$. |
89 | 469 |
Let $\phi_1, \phi_2, \ldots$ be an increasing sequence of real numbers satisfying |
470 |
$\phi_1 \ge 2$ and $\phi_{i+1} \ge \phi_i(2\phi_i + 2) + \phi_i$. |
|
471 |
For convenience, let $\phi_0 = 0$. |
|
248 | 472 |
Assume also that $\phi_k r \le \rho(M)$, |
473 |
where $\rho(M)$ is as in Lemma \ref{xxzz11}. |
|
89 | 474 |
Then there exists a neighborhood $U$ of $S$, |
475 |
homeomorphic to a disjoint union of balls, such that |
|
86 | 476 |
\[ |
89 | 477 |
\Nbd_{\phi_{k-1} r}(S) \subeq U \subeq \Nbd_{\phi_k r}(S) . |
86 | 478 |
\] |
479 |
\end{lemma} |
|
480 |
||
481 |
\begin{proof} |
|
89 | 482 |
For $k=1$ this follows from Lemma \ref{xxzz11}. |
483 |
Assume inductively that it holds for $k-1$. |
|
86 | 484 |
Partition $S$ into $k$ disjoint subsets $S_1,\ldots,S_k$, each of radius $\le r$. |
89 | 485 |
By Lemma \ref{xxzz11}, each $\Nbd_{\phi_{k-1} r}(S_i)$ is homeomorphic to a ball. |
486 |
If these balls are disjoint, let $U$ be their union. |
|
487 |
Otherwise, assume WLOG that $S_{k-1}$ and $S_k$ are distance less than $2\phi_{k-1}r$ apart. |
|
488 |
Let $R_i = \Nbd_{\phi_{k-1} r}(S_i)$ for $i = 1,\ldots,k-2$ |
|
489 |
and $R_{k-1} = \Nbd_{\phi_{k-1} r}(S_{k-1})\cup \Nbd_{\phi_{k-1} r}(S_k)$. |
|
490 |
Each $R_i$ is contained in a metric ball of radius $r' \deq (2\phi_{k-1}+2)r$. |
|
91 | 491 |
Note that the defining inequality of the $\phi_i$ guarantees that |
492 |
\[ |
|
493 |
\phi_{k-1}r' = \phi_{k-1}(2\phi_{k-1}+2)r \le \phi_k r \le \rho(M) . |
|
494 |
\] |
|
89 | 495 |
By induction, there is a neighborhood $U$ of $R \deq \bigcup_i R_i$, |
496 |
homeomorphic to a disjoint union |
|
497 |
of balls, and such that |
|
86 | 498 |
\[ |
89 | 499 |
U \subeq \Nbd_{\phi_{k-1}r'}(R) = \Nbd_{t}(S) \subeq \Nbd_{\phi_k r}(S) , |
86 | 500 |
\] |
89 | 501 |
where $t = \phi_{k-1}(2\phi_{k-1}+2)r + \phi_{k-1} r$. |
86 | 502 |
\end{proof} |
503 |
||
430 | 504 |
|
505 |
We now return to defining the chain maps $e_X$. |
|
506 |
||
70 | 507 |
|
254 | 508 |
Let $R_*$ be the chain complex with a generating 0-chain for each non-negative |
509 |
integer and a generating 1-chain connecting each adjacent pair $(j, j+1)$. |
|
358
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|
510 |
(So $R_*$ is a simplicial version of the non-negative reals.) |
254 | 511 |
Denote the 0-chains by $j$ (for $j$ a non-negative integer) and the 1-chain connecting $j$ and $j+1$ |
512 |
by $\iota_j$. |
|
513 |
Define a map (homotopy equivalence) |
|
250 | 514 |
\[ |
254 | 515 |
\sigma: R_*\ot CH_*(X, X) \otimes \bc_*(X) \to CH_*(X, X)\ot \bc_*(X) |
250 | 516 |
\] |
254 | 517 |
as follows. |
518 |
On $R_0\ot CH_*(X, X) \otimes \bc_*(X)$ we define |
|
519 |
\[ |
|
520 |
\sigma(j\ot p\ot b) = g_j(p)\ot b . |
|
521 |
\] |
|
255 | 522 |
On $R_1\ot CH_*(X, X) \otimes \bc_*(X)$ we define |
523 |
\[ |
|
524 |
\sigma(\iota_j\ot p\ot b) = f_j(p)\ot b , |
|
525 |
\] |
|
526 |
where $f_j$ is the homotopy from $g_j$ to $g_{j+1}$. |
|
86 | 527 |
|
254 | 528 |
Next we specify subcomplexes $G^m_* \sub R_*\ot CH_*(X, X) \otimes \bc_*(X)$ on which we will eventually |
529 |
define a version of the action map $e_X$. |
|
255 | 530 |
A generator $j\ot p\ot b$ is defined to be in $G^m_*$ if $j\ge j_{kbmp}$, where |
254 | 531 |
$k = k_{bmp}$ is the constant from Lemma \ref{Gim_approx}. |
255 | 532 |
Similarly $\iota_j\ot p\ot b$ is in $G^m_*$ if $j\ge j_{kbmp}$. |
254 | 533 |
The inequality following Lemma \ref{Gim_approx} guarantees that $G^m_*$ is indeed a subcomplex |
534 |
and that $G^m_* \sup G^{m+1}_*$. |
|
250 | 535 |
|
254 | 536 |
It is easy to see that each $G^m_*$ is homotopy equivalent (via the inclusion map) |
537 |
to $R_*\ot CH_*(X, X) \otimes \bc_*(X)$ |
|
538 |
and hence to $CH_*(X, X) \otimes \bc_*(X)$, and furthermore that the homotopies are well-defined |
|
539 |
up to a contractible set of choices. |
|
250 | 540 |
|
254 | 541 |
Next we define a map |
542 |
\[ |
|
543 |
e_m : G^m_* \to \bc_*(X) . |
|
544 |
\] |
|
255 | 545 |
Let $p\ot b$ be a generator of $G^m_*$. |
546 |
Each $g_j(p)\ot b$ or $f_j(p)\ot b$ is a linear combination of generators $q\ot c$, |
|
547 |
where $\supp(q)\cup\supp(c)$ is contained in a disjoint union of balls satisfying |
|
548 |
various conditions specified above. |
|
549 |
As in the construction of the maps $e_{i,m}$ above, |
|
550 |
it suffices to specify for each such $q\ot c$ a disjoint union of balls |
|
551 |
$V_{qc} \sup \supp(q)\cup\supp(c)$, such that $V_{qc} \sup V_{q'c'}$ |
|
552 |
whenever $q'\ot c'$ appears in the boundary of $q\ot c$. |
|
553 |
||
554 |
Let $q\ot c$ be a summand of $g_j(p)\ot b$, as above. |
|
555 |
Let $i$ be maximal such that $j\ge j_{ibmp}$ |
|
556 |
(notation as in Lemma \ref{Gim_approx}). |
|
557 |
Then $q\ot c \in G^{i,m}_*$ and we choose $V_{qc} \sup \supp(q)\cup\supp(c)$ |
|
558 |
such that |
|
559 |
\[ |
|
560 |
N_{i,d}(q\ot c) \subeq V_{qc} \subeq N_{i,d+1}(q\ot c) , |
|
561 |
\] |
|
562 |
where $d = \deg(q\ot c)$. |
|
563 |
Let $\tilde q = f_j(q)$. |
|
564 |
The summands of $f_j(p)\ot b$ have the form $\tilde q \ot c$, |
|
565 |
where $q\ot c$ is a summand of $g_j(p)\ot b$. |
|
566 |
Since the homotopy $f_j$ does not increase supports, we also have that |
|
567 |
\[ |
|
568 |
V_{qc} \sup \supp(\tilde q) \cup \supp(c) . |
|
569 |
\] |
|
570 |
So we define $V_{\tilde qc} = V_{qc}$. |
|
571 |
||
572 |
It is now easy to check that we have $V_{qc} \sup V_{q'c'}$ |
|
573 |
whenever $q'\ot c'$ appears in the boundary of $q\ot c$. |
|
574 |
As in the construction of the maps $e_{i,m}$ above, |
|
575 |
this allows us to construct a map |
|
576 |
\[ |
|
577 |
e_m : G^m_* \to \bc_*(X) |
|
578 |
\] |
|
579 |
which is well-defined up to homotopy. |
|
580 |
As in the proof of Lemma \ref{m_order_hty}, we can show that the map is well-defined up |
|
581 |
to $m$-th order homotopy. |
|
582 |
Put another way, we have specified an $m$-connected subcomplex of the complex of |
|
583 |
all maps $G^m_* \to \bc_*(X)$. |
|
584 |
On $G^{m+1}_* \sub G^m_*$ we have defined two maps, $e_m$ and $e_{m+1}$. |
|
585 |
One can similarly (to the proof of Lemma \ref{m_order_hty}) show that |
|
586 |
these two maps agree up to $m$-th order homotopy. |
|
587 |
More precisely, one can show that the subcomplex of maps containing the various |
|
588 |
$e_{m+1}$ candidates is contained in the corresponding subcomplex for $e_m$. |
|
253
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589 |
|
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|
590 |
\medskip |
8589275ac65b
CH_* action -- gluing compatibility
Kevin Walker <kevin@canyon23.net>
parents:
357
diff
changeset
|
591 |
|
8589275ac65b
CH_* action -- gluing compatibility
Kevin Walker <kevin@canyon23.net>
parents:
357
diff
changeset
|
592 |
Next we show that the action maps are compatible with gluing. |
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CH_* action -- gluing compatibility
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parents:
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diff
changeset
|
593 |
Let $G^m_*$ and $\ol{G}^m_*$ be the complexes, as above, used for defining |
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594 |
the action maps $e_{X\sgl}$ and $e_X$. |
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595 |
The gluing map $X\sgl\to X$ induces a map |
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596 |
\[ |
430 | 597 |
\gl: R_*\ot CH_*(X, X) \otimes \bc_*(X) \to R_*\ot CH_*(X\sgl, X \sgl) \otimes \bc_*(X \sgl) , |
358
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598 |
\] |
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599 |
and it is easy to see that $\gl(G^m_*)\sub \ol{G}^m_*$. |
437 | 600 |
From this it follows that the diagram in the statement of Theorem \ref{thm:CH} commutes. |
358
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601 |
|
430 | 602 |
\todo{this paragraph isn't very convincing, or at least I don't see what's going on} |
358
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603 |
Finally we show that the action maps defined above are independent of |
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604 |
the choice of metric (up to iterated homotopy). |
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605 |
The arguments are very similar to ones given above, so we only sketch them. |
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606 |
Let $g$ and $g'$ be two metrics on $X$, and let $e$ and $e'$ be the corresponding |
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607 |
actions $CH_*(X, X) \ot \bc_*(X)\to\bc_*(X)$. |
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608 |
We must show that $e$ and $e'$ are homotopic. |
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609 |
As outlined in the discussion preceding this proof, |
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610 |
this follows from the facts that both $e$ and $e'$ are compatible |
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611 |
with gluing and that $\bc_*(B^n)$ is contractible. |
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612 |
As above, we define a subcomplex $F_*\sub CH_*(X, X) \ot \bc_*(X)$ generated |
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613 |
by $p\ot b$ such that $|p|\cup|b|$ is contained in a disjoint union of balls. |
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614 |
Using acyclic models, we can construct a homotopy from $e$ to $e'$ on $F_*$. |
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615 |
We now observe that $CH_*(X, X) \ot \bc_*(X)$ retracts to $F_*$. |
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616 |
Similar arguments show that this homotopy from $e$ to $e'$ is well-defined |
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617 |
up to second order homotopy, and so on. |
430 | 618 |
|
437 | 619 |
This completes the proof of Theorem \ref{thm:CH}. |
84 | 620 |
\end{proof} |
621 |
||
622 |
||
396
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623 |
\begin{rem*} |
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624 |
\label{rem:for-small-blobs} |
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625 |
For the proof of Lemma \ref{lem:CH-small-blobs} below we will need the following observation on the action constructed above. |
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626 |
Let $b$ be a blob diagram and $p:P\times X\to X$ be a family of homeomorphisms. |
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627 |
Then we may choose $e$ such that $e(p\ot b)$ is a sum of generators, each |
385
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628 |
of which has support close to $p(t,|b|)$ for some $t\in P$. |
430 | 629 |
More precisely, the support of the generators is contained in the union of a small neighborhood |
630 |
of $p(t,|b|)$ with some small balls. |
|
385
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631 |
(Here ``small" is in terms of the metric on $X$ that we chose to construct $e$.) |
396
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632 |
\end{rem*} |
385
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633 |
|
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634 |
|
437 | 635 |
\begin{thm} |
636 |
\label{thm:CH-associativity} |
|
357
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637 |
The $CH_*(X, Y)$ actions defined above are associative. |
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638 |
That is, the following diagram commutes up to homotopy: |
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|
639 |
\[ \xymatrix{ |
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640 |
& CH_*(Y, Z) \ot \bc_*(Y) \ar[dr]^{e_{YZ}} & \\ |
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641 |
CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & \bc_*(Z) \\ |
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642 |
& CH_*(X, Z) \ot \bc_*(X) \ar[ur]_{e_{XZ}} & |
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643 |
} \] |
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644 |
Here $\mu:CH_*(X, Y) \ot CH_*(Y, Z)\to CH_*(X, Z)$ is the map induced by composition |
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645 |
of homeomorphisms. |
437 | 646 |
\end{thm} |
70 | 647 |
|
357
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648 |
\begin{proof} |
437 | 649 |
The strategy of the proof is similar to that of Theorem \ref{thm:CH}. |
357
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|
650 |
We will identify a subcomplex |
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651 |
\[ |
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|
652 |
G_* \sub CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) |
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653 |
\] |
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654 |
where it is easy to see that the two sides of the diagram are homotopic, then |
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|
655 |
show that there is a deformation retraction of $CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)$ into $G_*$. |
70 | 656 |
|
357
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657 |
Let $p\ot q\ot b$ be a generator of $CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)$. |
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|
658 |
By definition, $p\ot q\ot b\in G_*$ if there is a disjoint union of balls in $X$ which |
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659 |
contains $|p| \cup p\inv(|q|) \cup |b|$. |
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660 |
(If $p:P\times X\to Y$, then $p\inv(|q|)$ means the union over all $x\in P$ of |
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661 |
$p(x, \cdot)\inv(|q|)$.) |
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662 |
|
437 | 663 |
As in the proof of Theorem \ref{thm:CH}, we can construct a homotopy |
357
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|
664 |
between the upper and lower maps restricted to $G_*$. |
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|
665 |
This uses the facts that the maps agree on $CH_0(X, Y) \ot CH_0(Y, Z) \ot \bc_*(X)$, |
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666 |
that they are compatible with gluing, and the contractibility of $\bc_*(X)$. |
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|
667 |
|
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|
668 |
We can now apply Lemma \ref{extension_lemma_c}, using a series of increasingly fine covers, |
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|
669 |
to construct a deformation retraction of $CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)$ into $G_*$. |
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|
670 |
\end{proof} |