# HG changeset patch # User kevin@6e1638ff-ae45-0410-89bd-df963105f760 # Date 1247379285 0 # Node ID 6c7662fcddc56eb84bfa37061b6affe1b01cc9d8 # Parent 014a16e6e55c9f9056f62e0cfce3df1907210240 ... diff -r 014a16e6e55c -r 6c7662fcddc5 text/evmap.tex --- a/text/evmap.tex Sat Jul 11 06:40:01 2009 +0000 +++ b/text/evmap.tex Sun Jul 12 06:14:45 2009 +0000 @@ -165,7 +165,7 @@ 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} (e.g. $\phi_l = 2^{3^{l-1}}$). +satisfying the inequalities of Lemma \ref{xx2phi}. Given a generator $p\otimes b$ of $CD_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $l$ define \[ @@ -231,7 +231,7 @@ We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$. %We also have that $\deg(b'') = 0 = \deg(p'')$. Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$. -This is possible by \ref{bcontract}, \ref{disjunion} and \nn{prop. 2 of local relations (isotopy)}. +This is possible by \ref{bcontract}, \ref{disjunion} and \nn{property 2 of local relations (isotopy)}. Finally, define \[ e(p\ot b) \deq x' \bullet p''(b'') . @@ -374,70 +374,71 @@ \begin{proof} \label{xxyy2} Let $S$ be contained in $B_r(y)$, $y \in \ebb^n$. -Note that $\Nbd_a(S) \sup B_r(y)$. -Simple applications of the triangle inequality show that $\Nbd_a(S)$ -is star-shaped with respect to $y$. +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 $s\in S$. +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: + +\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 $2r \le a \le \rho(M)$, +$\Nbd_a(S)$ is homeomorphic to a ball. +\end{lemma} + +\begin{proof} +Choose $\rho = \rho(M)$ such that $3\rho/2$ is less than the radius of injectivity of $M$, +and also so that for any point $y\in M$ the geodesic coordinates of radius $3\rho/2$ around +$y$ distort angles by only a small amount. +Now the argument of the previous lemma works. \end{proof} -\begin{lemma} \label{xxyy3} -Let $S \sub \ebb^n$ be contained in a union (not necessarily disjoint) + +\begin{lemma} \label{xx2phi} +Let $S \sub M$ be contained in a union (not necessarily disjoint) of $k$ metric balls of radius $r$. -Then there exists a neighborhood $U$ of $S$ such that $U$ is homeomorphic to a disjoint union -of balls and +Let $\phi_1, \phi_2, \ldots$ be an increasing sequence of real numbers satisfying +$\phi_1 \ge 2$ and $\phi_{i+1} \ge \phi_i(2\phi_i + 2) + \phi_i$. +For convenience, let $\phi_0 = 0$. +Assume also that $\phi_k r \le \rho(M)$. +Then there exists a neighborhood $U$ of $S$, +homeomorphic to a disjoint union of balls, such that \[ - \Nbd_{2r}(S) \subeq U \subeq \Nbd_{4^k r}(S) . + \Nbd_{\phi_{k-1} r}(S) \subeq U \subeq \Nbd_{\phi_k r}(S) . \] \end{lemma} \begin{proof} +For $k=1$ this follows from Lemma \ref{xxzz11}. +Assume inductively that it holds for $k-1$. Partition $S$ into $k$ disjoint subsets $S_1,\ldots,S_k$, each of radius $\le r$. -By Lemma \ref{xxyy2}, each $\Nbd(S_i)$ is homeomorphic to a ball. -If these balls are disjoint (always the case if $k=1$) we are done. -If two (or more) of them intersect, then $S$ is contained in a union of $k-1$ metric -balls of radius $4r$. -By induction, there is a neighborhood $U$ of $S$ such that +By Lemma \ref{xxzz11}, each $\Nbd_{\phi_{k-1} r}(S_i)$ is homeomorphic to a ball. +If these balls are disjoint, let $U$ be their union. +Otherwise, assume WLOG that $S_{k-1}$ and $S_k$ are distance less than $2\phi_{k-1}r$ apart. +Let $R_i = \Nbd_{\phi_{k-1} r}(S_i)$ for $i = 1,\ldots,k-2$ +and $R_{k-1} = \Nbd_{\phi_{k-1} r}(S_{k-1})\cup \Nbd_{\phi_{k-1} r}(S_k)$. +Each $R_i$ is contained in a metric ball of radius $r' \deq (2\phi_{k-1}+2)r$. +By induction, there is a neighborhood $U$ of $R \deq \bigcup_i R_i$, +homeomorphic to a disjoint union +of balls, and such that \[ - U \subeq \Nbd_{4^{k-1}\cdot4r} . + U \subeq \Nbd_{\phi_{k-1}r'}(R) = \Nbd_{t}(S) \subeq \Nbd_{\phi_k r}(S) , \] +where $t = \phi_{k-1}(2\phi_{k-1}+2)r + \phi_{k-1} r$. \end{proof} -\begin{lemma} \label{xxyy4} -Let $S \sub \ebb^n$ be contained in a union (not necessarily disjoint) -of $k$ metric balls of radius $r$. -Then there exist neighborhoods $U_0, U_1, U_2, \ldots$ of $S$, -each homeomorphic to a disjoint union of balls, such that -\[ - \Nbd_{2r}(S) \subeq U_0 \subeq \Nbd_{4^k r}(S) - \subeq U_1 \subeq \Nbd_{4^{2k} r}(S) - \subeq U_2 \subeq \Nbd_{4^{3k} r}(S) \cdots -\] -\end{lemma} -\begin{proof} -Apply Lemma \ref {xxyy3} repeatedly. -\end{proof} - -\begin{lemma} \label{xxyy5} -Let $M$ be a Riemannian $n$-manifold and positive integers $m$ and $k$. -There exists a constant $\eta(M, m, k)$ such that for all subsets -$S\subeq M$ which are contained in a (not necessarily disjoint) union of -$k$ metric balls of radius $r$, $r < \eta(M, m, k)$, -there exist neighborhoods $U_0, U_1, \ldots, U_m$ of $S$, -each homeomorphic to a disjoint union of balls, such that -\[ - \Nbd_{2r}(S) \subeq U_0 \subeq \Nbd_{4^k r}(S) - \subeq U_1 \subeq \Nbd_{4^{2k} r}(S) \cdots - \subeq U_m \subeq \Nbd_{4^{(m+1)k} r}(S) . -\] - -\end{lemma} - -\begin{proof} -Choose $\eta = \eta(M, m, k)$ small enough so that metric balls of radius $4^{(m+1)k} \eta$ -are injective and also have small distortion with respect to a Euclidean metric. -Then proceed as in the proof of Lemma \ref{xxyy4}. -\end{proof} \medskip