author | Kevin Walker <kevin@canyon23.net> |
Tue, 06 Apr 2010 08:43:37 -0700 | |
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\section{Action of \texorpdfstring{$\CH{X}$}{$C_*(Homeo(M))$}} |
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\label{sec:evaluation} |
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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 $Y$ (extending a fixed homeomorphism $\bd X \to \bd Y$). |
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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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We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
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\begin{prop} \label{CHprop} |
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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 $\bc_*(X)$ (Proposition (\ref{diff0prop})), 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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\eq{ \xymatrix{ |
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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} \\ |
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CH_*(X, Y) \otimes \bc_*(X) |
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\ar@/_4ex/[r]_{e_{XY}} & |
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\bc_*(Y) |
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} } |
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\end{enumerate} |
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Up to (iterated) homotopy, there is a unique family $\{e_{XY}\}$ of chain maps |
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satisfying the above two conditions. |
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\end{prop} |
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\nn{Also need to say something about associativity. |
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Put it in the above prop or make it a separate prop? |
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I lean toward the latter.} |
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\medskip |
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The proof will occupy the the next several pages. |
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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 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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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 $\cU$} if there is a factorization |
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\eq{ |
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P = P_1 \times \cdots \times P_m |
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} |
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(for some $m \le k$) |
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and families of homeomorphisms |
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\eq{ |
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f_i : P_i \times X \to X |
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} |
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such that |
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\begin{itemize} |
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\item each $f_i$ is supported on some connected $V_i \sub X$; |
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\item the sets $V_i$ are mutually disjoint; |
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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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where $k_i = \dim(P_i)$; and |
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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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for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Homeo(X)$. |
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\end{itemize} |
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A chain $x \in CH_k(X)$ is (by definition) adapted to $\cU$ if it is the sum |
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of singular cells, each of which is adapted to $\cU$. |
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(Actually, in this section we will only need families of homeomorphisms to be |
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{\it weakly adapted} to $\cU$, meaning that 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 Section \ref{sec:localising}. |
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We will actually prove the following more general result. |
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Let $S$ and $T$ be an arbitrary topological spaces. |
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%\nn{might need to restrict $S$; the proof uses partition of unity on $S$; |
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%check this; or maybe just restrict the cover} |
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Let $CM_*(S, T)$ denote the singular chains on the space of continuous maps |
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from $S$ to $T$. |
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Let $\cU$ be an open cover of $S$ which affords a partition of unity. |
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\nn{for some $S$ and $\cU$ there is no partition of unity? like if $S$ is not paracompact? |
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in any case, in our applications $S$ will always be a manifold} |
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\begin{lemma} \label{extension_lemma_b} |
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Let $x \in CM_k(S, T)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
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Then $x$ is homotopic (rel boundary) to some $x' \in CM_k(S, T)$ 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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If $S$ and $T$ are manifolds, the statement remains true if we replace $CM_*(S, T)$ with |
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chains of smooth maps or immersions. |
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\end{lemma} |
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\medskip |
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Before diving into the details, we outline our strategy for the proof of Proposition \ref{CHprop}. |
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%Suppose for the moment that evaluation maps with the advertised properties exist. |
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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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Suppose that there exists $V \sub X$ such that |
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\begin{enumerate} |
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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{enumerate} |
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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 plain, 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 \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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\[ |
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\bd(e_{VV'}(q\otimes b_V)) = e_{VV'}(\bd(q\otimes b_V)) . |
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\] |
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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, etc |
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We expect that everything is true in the PL category, but at the moment our proof |
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avails itself to smooth techniques. |
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Furthermore, it traditional in the literature to speak of $C_*(\Diff(X))$ |
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rather than $C_*(\Homeo(X))$.} |
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\medskip |
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Now for the details. |
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Notation: Let $|b| = \supp(b)$, $|p| = \supp(p)$. |
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Choose a metric on $X$. |
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Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero |
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(e.g.\ $\ep_i = 2^{-i}$). |
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Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$ |
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converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$). |
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Let $\phi_l$ be an increasing sequence of positive numbers |
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satisfying the inequalities of Lemma \ref{xx2phi}. |
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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, 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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$p\ot b$ is (by definition) in $G_*^{i,m}$ if either (a) $\deg(p) = 0$ or (b) |
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there exist codimension-zero submanifolds $V_0,\ldots,V_m \sub X$ 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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Further, we require (inductively) that $\bd(p\ot b) \in G_*^{i,m}$. |
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We also require that $b$ is splitable (transverse) along the boundary of each $V_l$. |
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Note that $G_*^{i,m+1} \subeq G_*^{i,m}$. |
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As sketched above and explained in detail below, |
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$G_*^{i,m}$ is a subcomplex where it is easy to define |
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the evaluation map. |
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The parameter $m$ controls the number of iterated homotopies we are able to construct |
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(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) $e: G_*^{i,m} \to \bc_*(X)$. |
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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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\[ |
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p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet b'' , |
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\] |
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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 \ref{bcontract}, \ref{disjunion} and \nn{property 2 of local relations (isotopy)}. |
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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. |
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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} |
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\begin{lemma} \label{m_order_hty} |
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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 $V$ (and hence also different choices of $x'$) at each step. |
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If $m \ge 1$ then $e$ and $\tilde{e}$ are homotopic. |
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If $m \ge 2$ then any two choices of this first-order homotopy are second-order homotopic. |
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And so on. |
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In other words, $e : G_*^{i,m} \to \bc_*(X)$ is well-defined up to $m$-th order homotopy. |
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\end{lemma} |
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\begin{proof} |
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We construct $h: G_*^{i,m} \to \bc_*(X)$ such that $\bd h + h\bd = e - \tilde{e}$. |
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$e$ and $\tilde{e}$ coincide on bidegrees $(0, j)$, so define $h$ |
|
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to be zero there. |
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Assume inductively that $h$ has been defined for degrees less than $k$. |
|
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Let $p\ot b$ be a generator of degree $k$. |
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Choose $V_1$ as in the definition of $G_*^{i,m}$ so that |
|
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\[ |
|
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N_{i,k+1}(p\ot b) \subeq V_1 \subeq N_{i,k+2}(p\ot b) . |
|
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\] |
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There are splittings |
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\[ |
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p = p'_1\bullet p''_1 , \;\; b = b'_1\bullet b''_1 , |
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\;\; e(p\ot b) - \tilde{e}(p\ot b) - h(\bd(p\ot b)) = f'_1\bullet f''_1 , |
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\] |
|
236 | 302 |
where $p'_1 \in CH_*(V_1)$, $p''_1 \in CH_*(X\setmin V_1)$, |
84 | 303 |
$b'_1 \in \bc_*(V_1)$, $b''_1 \in \bc_*(X\setmin V_1)$, |
304 |
$f'_1 \in \bc_*(p(V_1))$, and $f''_1 \in \bc_*(p(X\setmin V_1))$. |
|
88 | 305 |
Inductively, $\bd f'_1 = 0$ and $f_1'' = p_1''(b_1'')$. |
84 | 306 |
Choose $x'_1 \in \bc_*(p(V_1))$ so that $\bd x'_1 = f'_1$. |
307 |
Define |
|
308 |
\[ |
|
309 |
h(p\ot b) \deq x'_1 \bullet p''_1(b''_1) . |
|
310 |
\] |
|
311 |
This completes the construction of the first-order homotopy when $m \ge 1$. |
|
312 |
||
313 |
The $j$-th order homotopy is constructed similarly, with $V_j$ replacing $V_1$ above. |
|
314 |
\end{proof} |
|
315 |
||
316 |
Note that on $G_*^{i,m+1} \subeq G_*^{i,m}$, we have defined two maps, |
|
317 |
call them $e_{i,m}$ and $e_{i,m+1}$. |
|
318 |
An easy variation on the above lemma shows that $e_{i,m}$ and $e_{i,m+1}$ are $m$-th |
|
319 |
order homotopic. |
|
320 |
||
236 | 321 |
Next we show how to homotope chains in $CH_*(X)\ot \bc_*(X)$ to one of the |
85 | 322 |
$G_*^{i,m}$. |
323 |
Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero. |
|
324 |
Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$. |
|
236 | 325 |
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}. |
86 | 326 |
Recall that $h_j$ and also the homotopy connecting it to the identity do not increase |
85 | 327 |
supports. |
328 |
Define |
|
329 |
\[ |
|
330 |
g_j \deq h_j\circ h_{j-1} \circ \cdots \circ h_1 . |
|
331 |
\] |
|
332 |
The next lemma says that for all generators $p\ot b$ we can choose $j$ large enough so that |
|
333 |
$g_j(p)\ot b$ lies in $G_*^{i,m}$, for arbitrary $m$ and sufficiently large $i$ |
|
334 |
(depending on $b$, $n = \deg(p)$ and $m$). |
|
88 | 335 |
(Note: Don't confuse this $n$ with the top dimension $n$ used elsewhere in this paper.) |
85 | 336 |
|
87 | 337 |
\begin{lemma} \label{Gim_approx} |
236 | 338 |
Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CH_*(X)$. |
85 | 339 |
Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$ |
236 | 340 |
there exists another constant $j_i$ such that for all $j \ge j_i$ and all $p\in CH_n(X)$ |
85 | 341 |
we have $g_j(p)\ot b \in G_*^{i,m}$. |
342 |
\end{lemma} |
|
343 |
||
344 |
\begin{proof} |
|
345 |
Let $c$ be a subset of the blobs of $b$. |
|
346 |
There exists $l > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < l$ |
|
347 |
and all such $c$. |
|
86 | 348 |
(Here we are using a piecewise smoothness assumption for $\bd c$, and also |
90 | 349 |
the fact that $\bd c$ is collared. |
350 |
We need to consider all such $c$ because all generators appearing in |
|
351 |
iterated boundaries of must be in $G_*^{i,m}$.) |
|
85 | 352 |
|
86 | 353 |
Let $r = \deg(b)$ and |
354 |
\[ |
|
90 | 355 |
t = r+n+m+1 = \deg(p\ot b) + m + 1. |
86 | 356 |
\] |
85 | 357 |
|
358 |
Choose $k = k_{bmn}$ such that |
|
359 |
\[ |
|
86 | 360 |
t\ep_k < l |
85 | 361 |
\] |
362 |
and |
|
363 |
\[ |
|
90 | 364 |
n\cdot (2 (\phi_t + 1) \delta_k) < \ep_k . |
85 | 365 |
\] |
366 |
Let $i \ge k_{bmn}$. |
|
367 |
Choose $j = j_i$ so that |
|
368 |
\[ |
|
90 | 369 |
\gamma_j < \delta_i |
370 |
\] |
|
371 |
and also so that $\phi_t \gamma_j$ is less than the constant $\rho(M)$ of Lemma \ref{xxzz11}. |
|
372 |
||
236 | 373 |
Let $j \ge j_i$ and $p\in CH_n(X)$. |
90 | 374 |
Let $q$ be a generator appearing in $g_j(p)$. |
375 |
Note that $|q|$ is contained in a union of $n$ elements of the cover $\cU_j$, |
|
376 |
which implies that $|q|$ is contained in a union of $n$ metric balls of radius $\delta_i$. |
|
377 |
We must show that $q\ot b \in G_*^{i,m}$, which means finding neighborhoods |
|
378 |
$V_0,\ldots,V_m \sub X$ of $|q|\cup |b|$ such that each $V_j$ |
|
379 |
is homeomorphic to a disjoint union of balls and |
|
380 |
\[ |
|
381 |
N_{i,n}(q\ot b) \subeq V_0 \subeq N_{i,n+1}(q\ot b) |
|
382 |
\subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,t}(q\ot b) . |
|
383 |
\] |
|
384 |
By repeated applications of Lemma \ref{xx2phi} we can find neighborhoods $U_0,\ldots,U_m$ |
|
385 |
of $|q|$, each homeomorphic to a disjoint union of balls, with |
|
386 |
\[ |
|
387 |
\Nbd_{\phi_{n+l} \delta_i}(|q|) \subeq U_l \subeq \Nbd_{\phi_{n+l+1} \delta_i}(|q|) . |
|
85 | 388 |
\] |
90 | 389 |
The inequalities above \nn{give ref} guarantee that we can find $u_l$ with |
390 |
\[ |
|
391 |
(n+l)\ep_i \le u_l \le (n+l+1)\ep_i |
|
392 |
\] |
|
393 |
such that each component of $U_l$ is either disjoint from $\Nbd_{u_l}(|b|)$ or contained in |
|
394 |
$\Nbd_{u_l}(|b|)$. |
|
395 |
This is because there are at most $n$ components of $U_l$, and each component |
|
396 |
has radius $\le (\phi_t + 1) \delta_i$. |
|
397 |
It follows that |
|
398 |
\[ |
|
399 |
V_l \deq \Nbd_{u_l}(|b|) \cup U_l |
|
400 |
\] |
|
401 |
is homeomorphic to a disjoint union of balls and satisfies |
|
402 |
\[ |
|
403 |
N_{i,n+l}(q\ot b) \subeq V_l \subeq N_{i,n+l+1}(q\ot b) . |
|
404 |
\] |
|
86 | 405 |
|
90 | 406 |
The same argument shows that each generator involved in iterated boundaries of $q\ot b$ |
407 |
is in $G_*^{i,m}$. |
|
86 | 408 |
\end{proof} |
409 |
||
410 |
In the next few lemmas we have made no effort to optimize the various bounds. |
|
411 |
(The bounds are, however, optimal in the sense of minimizing the amount of work |
|
412 |
we do. Equivalently, they are the first bounds we thought of.) |
|
413 |
||
414 |
We say that a subset $S$ of a metric space has radius $\le r$ if $S$ is contained in |
|
415 |
some metric ball of radius $r$. |
|
416 |
||
417 |
\begin{lemma} |
|
418 |
Let $S \sub \ebb^n$ (Euclidean $n$-space) have radius $\le r$. |
|
419 |
Then $\Nbd_a(S)$ is homeomorphic to a ball for $a \ge 2r$. |
|
420 |
\end{lemma} |
|
421 |
||
422 |
\begin{proof} \label{xxyy2} |
|
423 |
Let $S$ be contained in $B_r(y)$, $y \in \ebb^n$. |
|
89 | 424 |
Note that if $a \ge 2r$ then $\Nbd_a(S) \sup B_r(y)$. |
425 |
Let $z\in \Nbd_a(S) \setmin B_r(y)$. |
|
426 |
Consider the triangle |
|
427 |
\nn{give figure?} with vertices $z$, $y$ and $s$ with $s\in S$. |
|
428 |
The length of the edge $yz$ is greater than $r$ which is greater |
|
429 |
than the length of the edge $ys$. |
|
430 |
It follows that the angle at $z$ is less than $\pi/2$ (less than $\pi/3$, in fact), |
|
431 |
which means that points on the edge $yz$ near $z$ are closer to $s$ than $z$ is, |
|
432 |
which implies that these points are also in $\Nbd_a(S)$. |
|
433 |
Hence $\Nbd_a(S)$ is star-shaped with respect to $y$. |
|
434 |
\end{proof} |
|
435 |
||
436 |
If we replace $\ebb^n$ above with an arbitrary compact Riemannian manifold $M$, |
|
437 |
the same result holds, so long as $a$ is not too large: |
|
438 |
||
439 |
\begin{lemma} \label{xxzz11} |
|
440 |
Let $M$ be a compact Riemannian manifold. |
|
441 |
Then there is a constant $\rho(M)$ such that for all |
|
442 |
subsets $S\sub M$ of radius $\le r$ and all $a$ such that $2r \le a \le \rho(M)$, |
|
443 |
$\Nbd_a(S)$ is homeomorphic to a ball. |
|
444 |
\end{lemma} |
|
445 |
||
446 |
\begin{proof} |
|
447 |
Choose $\rho = \rho(M)$ such that $3\rho/2$ is less than the radius of injectivity of $M$, |
|
448 |
and also so that for any point $y\in M$ the geodesic coordinates of radius $3\rho/2$ around |
|
449 |
$y$ distort angles by only a small amount. |
|
450 |
Now the argument of the previous lemma works. |
|
85 | 451 |
\end{proof} |
452 |
||
453 |
||
89 | 454 |
|
455 |
\begin{lemma} \label{xx2phi} |
|
456 |
Let $S \sub M$ be contained in a union (not necessarily disjoint) |
|
86 | 457 |
of $k$ metric balls of radius $r$. |
89 | 458 |
Let $\phi_1, \phi_2, \ldots$ be an increasing sequence of real numbers satisfying |
459 |
$\phi_1 \ge 2$ and $\phi_{i+1} \ge \phi_i(2\phi_i + 2) + \phi_i$. |
|
460 |
For convenience, let $\phi_0 = 0$. |
|
461 |
Assume also that $\phi_k r \le \rho(M)$. |
|
462 |
Then there exists a neighborhood $U$ of $S$, |
|
463 |
homeomorphic to a disjoint union of balls, such that |
|
86 | 464 |
\[ |
89 | 465 |
\Nbd_{\phi_{k-1} r}(S) \subeq U \subeq \Nbd_{\phi_k r}(S) . |
86 | 466 |
\] |
467 |
\end{lemma} |
|
468 |
||
469 |
\begin{proof} |
|
89 | 470 |
For $k=1$ this follows from Lemma \ref{xxzz11}. |
471 |
Assume inductively that it holds for $k-1$. |
|
86 | 472 |
Partition $S$ into $k$ disjoint subsets $S_1,\ldots,S_k$, each of radius $\le r$. |
89 | 473 |
By Lemma \ref{xxzz11}, each $\Nbd_{\phi_{k-1} r}(S_i)$ is homeomorphic to a ball. |
474 |
If these balls are disjoint, let $U$ be their union. |
|
475 |
Otherwise, assume WLOG that $S_{k-1}$ and $S_k$ are distance less than $2\phi_{k-1}r$ apart. |
|
476 |
Let $R_i = \Nbd_{\phi_{k-1} r}(S_i)$ for $i = 1,\ldots,k-2$ |
|
477 |
and $R_{k-1} = \Nbd_{\phi_{k-1} r}(S_{k-1})\cup \Nbd_{\phi_{k-1} r}(S_k)$. |
|
478 |
Each $R_i$ is contained in a metric ball of radius $r' \deq (2\phi_{k-1}+2)r$. |
|
91 | 479 |
Note that the defining inequality of the $\phi_i$ guarantees that |
480 |
\[ |
|
481 |
\phi_{k-1}r' = \phi_{k-1}(2\phi_{k-1}+2)r \le \phi_k r \le \rho(M) . |
|
482 |
\] |
|
89 | 483 |
By induction, there is a neighborhood $U$ of $R \deq \bigcup_i R_i$, |
484 |
homeomorphic to a disjoint union |
|
485 |
of balls, and such that |
|
86 | 486 |
\[ |
89 | 487 |
U \subeq \Nbd_{\phi_{k-1}r'}(R) = \Nbd_{t}(S) \subeq \Nbd_{\phi_k r}(S) , |
86 | 488 |
\] |
89 | 489 |
where $t = \phi_{k-1}(2\phi_{k-1}+2)r + \phi_{k-1} r$. |
86 | 490 |
\end{proof} |
491 |
||
492 |
||
70 | 493 |
\medskip |
494 |
||
86 | 495 |
|
92 | 496 |
\hrule\medskip\hrule\medskip |
497 |
||
498 |
\nn{outline of what remains to be done:} |
|
499 |
||
500 |
\begin{itemize} |
|
501 |
\item We need to assemble the maps for the various $G^{i,m}$ into |
|
236 | 502 |
a map for all of $CH_*\ot \bc_*$. |
503 |
One idea: Think of the $g_j$ as a sort of homotopy (from $CH_*\ot \bc_*$ to itself) |
|
504 |
parameterized by $[0,\infty)$. For each $p\ot b$ in $CH_*\ot \bc_*$ choose a sufficiently |
|
92 | 505 |
large $j'$. Use these choices to reparameterize $g_\bullet$ so that each |
506 |
$p\ot b$ gets pushed as far as the corresponding $j'$. |
|
507 |
\item Independence of metric, $\ep_i$, $\delta_i$: |
|
508 |
For a different metric etc. let $\hat{G}^{i,m}$ denote the alternate subcomplexes |
|
509 |
and $\hat{N}_{i,l}$ the alternate neighborhoods. |
|
510 |
Main idea is that for all $i$ there exists sufficiently large $k$ such that |
|
511 |
$\hat{N}_{k,l} \sub N_{i,l}$, and similarly with the roles of $N$ and $\hat{N}$ reversed. |
|
512 |
\item Also need to prove associativity. |
|
513 |
\end{itemize} |
|
86 | 514 |
|
515 |
||
92 | 516 |
\nn{to be continued....} |
86 | 517 |
|
84 | 518 |
\noop{ |
519 |
||
520 |
\begin{lemma} |
|
521 |
||
522 |
\end{lemma} |
|
86 | 523 |
|
84 | 524 |
\begin{proof} |
525 |
||
526 |
\end{proof} |
|
527 |
||
528 |
} |
|
529 |
||
530 |
||
70 | 531 |
|
532 |
||
533 |
%\nn{say something about associativity here} |
|
534 |
||
535 |
||
536 |
||
537 |
||
538 |