120 To show existence, we must show that the various choices involved in constructing |
120 To show existence, we must show that the various choices involved in constructing |
121 evaluation maps in this way affect the final answer only by a homotopy. |
121 evaluation maps in this way affect the final answer only by a homotopy. |
122 |
122 |
123 Now for a little more detail. |
123 Now for a little more detail. |
124 (But we're still just motivating the full, gory details, which will follow.) |
124 (But we're still just motivating the full, gory details, which will follow.) |
125 Choose a metric on $X$, and let $\cU_\gamma$ be the open cover of balls of radius $\gamma$. |
125 Choose a metric on $X$, and let $\cU_\gamma$ be the open cover of $X$ by balls of radius $\gamma$. |
126 By Lemma \ref{extension_lemma} we can restrict our attention to $k$-parameter families |
126 By Lemma \ref{extension_lemma} we can restrict our attention to $k$-parameter families |
127 $p$ of homeomorphisms such that $\supp(p)$ is contained in the union of $k$ $\gamma$-balls. |
127 $p$ of homeomorphisms such that $\supp(p)$ is contained in the union of $k$ $\gamma$-balls. |
128 For fixed blob diagram $b$ and fixed $k$, it's not hard to show that for $\gamma$ small enough |
128 For fixed blob diagram $b$ and fixed $k$, it's not hard to show that for $\gamma$ small enough |
129 $p\ot b$ must be localizable. |
129 $p\ot b$ must be localizable. |
130 On the other hand, for fixed $k$ and $\gamma$ there exist $p$ and $b$ such that $p\ot b$ is not localizable, |
130 On the other hand, for fixed $k$ and $\gamma$ there exist $p$ and $b$ such that $p\ot b$ is not localizable, |
151 |
151 |
152 \begin{proof}[Proof of Proposition \ref{CHprop}.] |
152 \begin{proof}[Proof of Proposition \ref{CHprop}.] |
153 We'll use the notation $|b| = \supp(b)$ and $|p| = \supp(p)$. |
153 We'll use the notation $|b| = \supp(b)$ and $|p| = \supp(p)$. |
154 |
154 |
155 Choose a metric on $X$. |
155 Choose a metric on $X$. |
156 Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging monotonically to zero |
156 Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero |
157 (e.g.\ $\ep_i = 2^{-i}$). |
157 (e.g.\ $\ep_i = 2^{-i}$). |
158 Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$ |
158 Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$ |
159 converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$). |
159 converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$). |
160 Let $\phi_l$ be an increasing sequence of positive numbers |
160 Let $\phi_l$ be an increasing sequence of positive numbers |
161 satisfying the inequalities of Lemma \ref{xx2phi} below. |
161 satisfying the inequalities of Lemma \ref{xx2phi} below. |
175 We say $p\ot b$ is in $G_*^{i,m}$ exactly when either (a) $\deg(p) = 0$ or (b) |
175 We say $p\ot b$ is in $G_*^{i,m}$ exactly when either (a) $\deg(p) = 0$ or (b) |
176 there exist codimension-zero submanifolds $V_0,\ldots,V_m \sub X$ such that each $V_j$ |
176 there exist codimension-zero submanifolds $V_0,\ldots,V_m \sub X$ such that each $V_j$ |
177 is homeomorphic to a disjoint union of balls and |
177 is homeomorphic to a disjoint union of balls and |
178 \[ |
178 \[ |
179 N_{i,k}(p\ot b) \subeq V_0 \subeq N_{i,k+1}(p\ot b) |
179 N_{i,k}(p\ot b) \subeq V_0 \subeq N_{i,k+1}(p\ot b) |
180 \subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,k+m+1}(p\ot b) . |
180 \subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,k+m+1}(p\ot b) , |
181 \] |
181 \] |
182 and further $\bd(p\ot b) \in G_*^{i,m}$. |
182 and further $\bd(p\ot b) \in G_*^{i,m}$. |
183 We also require that $b$ is splitable (transverse) along the boundary of each $V_l$. |
183 We also require that $b$ is splitable (transverse) along the boundary of each $V_l$. |
184 |
184 |
185 Note that $G_*^{i,m+1} \subeq G_*^{i,m}$. |
185 Note that $G_*^{i,m+1} \subeq G_*^{i,m}$. |
343 |
343 |
344 |
344 |
345 \begin{proof} |
345 \begin{proof} |
346 |
346 |
347 There exists $\lambda > 0$ such that for every subset $c$ of the blobs of $b$ $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ . |
347 There exists $\lambda > 0$ such that for every subset $c$ of the blobs of $b$ $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ . |
348 (Here we are using the fact that the blobs are piecewise-linear and thatthat $\bd c$ is collared.) |
348 (Here we are using the fact that the blobs are |
|
349 piecewise smooth or piecewise-linear and that $\bd c$ is collared.) |
349 We need to consider all such $c$ because all generators appearing in |
350 We need to consider all such $c$ because all generators appearing in |
350 iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.) |
351 iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.) |
351 |
352 |
352 Let $r = \deg(b)$ and |
353 Let $r = \deg(b)$ and |
353 \[ |
354 \[ |