126 where $r(b_W)$ denotes the obvious action of diffeomorphisms on blob diagrams (in |
126 where $r(b_W)$ denotes the obvious action of diffeomorphisms on blob diagrams (in |
127 this case a 0-blob diagram). |
127 this case a 0-blob diagram). |
128 Since $V'$ is a disjoint union of balls, $\bc_*(V')$ is acyclic in degrees $>0$ |
128 Since $V'$ is a disjoint union of balls, $\bc_*(V')$ is acyclic in degrees $>0$ |
129 (by \ref{disjunion} and \ref{bcontract}). |
129 (by \ref{disjunion} and \ref{bcontract}). |
130 Assuming inductively that we have already defined $e_{VV'}(\bd(q\otimes b_V))$, |
130 Assuming inductively that we have already defined $e_{VV'}(\bd(q\otimes b_V))$, |
131 there is, up to homotopy, a unique choice for $e_{VV'}(q\otimes b_V)$ |
131 there is, up to (iterated) homotopy, a unique choice for $e_{VV'}(q\otimes b_V)$ |
132 such that |
132 such that |
133 \[ |
133 \[ |
134 \bd(e_{VV'}(q\otimes b_V)) = e_{VV'}(\bd(q\otimes b_V)) . |
134 \bd(e_{VV'}(q\otimes b_V)) = e_{VV'}(\bd(q\otimes b_V)) . |
135 \] |
135 \] |
136 |
136 |
137 Thus the conditions of the proposition determine (up to homotopy) the evaluation |
137 Thus the conditions of the proposition determine (up to homotopy) the evaluation |
138 map for generators $p\otimes b$ such that $\supp(p) \cup \supp(b)$ is contained in a disjoint |
138 map for generators $p\otimes b$ such that $\supp(p) \cup \supp(b)$ is contained in a disjoint |
139 union of balls. |
139 union of balls. |
140 On the other hand, Lemma \ref{extension_lemma} allows us to homotope |
140 On the other hand, Lemma \ref{extension_lemma} allows us to homotope |
141 \nn{is this commonly used as a verb?} arbitrary generators to sums of generators with this property. |
141 \nn{is this commonly used as a verb?} arbitrary generators to sums of generators with this property. |
142 \nn{should give a name to this property} |
142 \nn{should give a name to this property; also forward reference} |
143 This (roughly) establishes the uniqueness part of the proposition. |
143 This (roughly) establishes the uniqueness part of the proposition. |
144 To show existence, we must show that the various choices involved in constructing |
144 To show existence, we must show that the various choices involved in constructing |
145 evaluation maps in this way affect the final answer only by a homotopy. |
145 evaluation maps in this way affect the final answer only by a homotopy. |
146 |
146 |
147 \nn{now for a more detailed outline of the proof...} |
147 \nn{maybe put a little more into the outline before diving into the details.} |
148 |
148 |
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149 \nn{Note: At the moment this section is very inconsistent with respect to PL versus smooth, |
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150 homeomorphism versus diffeomorphism, etc. |
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151 We expect that everything is true in the PL category, but at the moment our proof |
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152 avails itself to smooth techniques. |
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153 Furthermore, it traditional in the literature to speak of $C_*(\Diff(X))$ |
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154 rather than $C_*(\Homeo(X))$.} |
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155 |
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156 \medskip |
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157 |
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158 Now for the details. |
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159 |
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160 Notation: Let $|b| = \supp(b)$, $|p| = \supp(p)$. |
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161 |
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162 Choose a metric on $X$. |
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163 Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero |
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164 (e.g.\ $\ep_i = 2^{-i}$). |
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165 Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$ |
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166 converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$). |
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167 Given a generator $p\otimes b$ of $CD_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $k$ |
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168 define |
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169 \[ |
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170 N_{i,k}(p\ot b) \deq \Nbd_{k\ep_i}(|b|) \cup \Nbd_{k\delta_i}(|p|). |
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171 \] |
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172 In other words, we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized |
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173 by $k$), with $\ep_i$ controlling the size of the buffer around $|b|$ and $\delta_i$ controlling |
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174 the size of the buffer around $|p|$. |
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175 |
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176 Next we define subcomplexes $G_*^{i,m} \sub CD_*(X)\otimes \bc_*(X)$. |
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177 Let $p\ot b$ be a generator of $CD_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b) |
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178 = \deg(p) + \deg(b)$. |
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179 $p\ot b$ is (by definition) in $G_*^{i,m}$ if either (a) $\deg(p) = 0$ or (b) |
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180 there exist codimension-zero submanifolds $V_1,\ldots,V_m \sub X$ such that each $V_j$ |
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181 is homeomorphic to a disjoint union of balls and |
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182 \[ |
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183 N_{i,k}(p\ot b) \subeq V_1 \subeq N_{i,k+1}(p\ot b) |
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184 \subeq V_2 \subeq \cdots \subeq V_m \subeq N_{i,k+m}(p\ot b) . |
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185 \] |
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186 Further, we require (inductively) that $\bd(p\ot b) \in G_*^{i,m}$. |
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187 We also require that $b$ is splitable (transverse) along the boundary of each $V_l$. |
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188 |
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189 Note that $G_*^{i,m+1} \subeq G_*^{i,m}$. |
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190 |
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191 As sketched above and explained in detail below, |
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192 $G_*^{i,m}$ is a subcomplex where it is easy to define |
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193 the evaluation map. |
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194 The parameter $m$ controls the number of iterated homotopies we are able to construct. |
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195 The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of |
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196 $CD_*(X)\ot \bc_*(X)$. |
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197 |
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198 Next we define a chain map (dependent on some choices) $e: G_*^{i,m} \to \bc_*(X)$. |
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199 Let $p\ot b \in G_*^{i,m}$. |
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200 If $\deg(p) = 0$, define |
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201 \[ |
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202 e(p\ot b) = p(b) , |
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203 \] |
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204 where $p(b)$ denotes the obvious action of the diffeomorphism(s) $p$ on the blob diagram $b$. |
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205 For general $p\ot b$ ($\deg(p) \ge 1$) assume inductively that we have already defined |
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206 $e(p'\ot b')$ when $\deg(p') + \deg(b') < k = \deg(p) + \deg(b)$. |
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207 Choose $V_1$ as above so that |
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208 \[ |
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209 N_{i,k}(p\ot b) \subeq V_1 \subeq N_{i,k+1}(p\ot b) . |
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210 \] |
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211 Let $\bd(p\ot b) = \sum_j p_j\ot b_j$, and let $V_1^j$ be the choice of neighborhood |
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212 of $|p_j|\cup |b_j|$ made at the preceding stage of the induction. |
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213 For all $j$, |
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214 \[ |
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215 V_1^j \subeq N_{i,(k-1)+1}(p_j\ot b_j) \subeq N_{i,k}(p\ot b) \subeq V_1 . |
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216 \] |
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217 (The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.) |
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218 We therefore have splittings |
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219 \[ |
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220 p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet f'' , |
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221 \] |
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222 where $p' \in CD_*(V_1)$, $p'' \in CD_*(X\setmin V_1)$, |
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223 $b' \in \bc_*(V_1)$, $b'' \in \bc_*(X\setmin V_1)$, |
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224 $e' \in \bc_*(p(V_1))$, and $e'' \in \bc_*(p(X\setmin V_1))$. |
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225 (Note that since the family of diffeomorphisms $p$ is constant (independent of parameters) |
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226 near $\bd V_1)$, the expressions $p(V_1) \sub X$ and $p(X\setmin V_1) \sub X$ are |
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227 unambiguous.) |
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228 We also have that $\deg(b'') = 0 = \deg(p'')$. |
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229 Choose $x' \in \bc_*(p(V_1))$ such that $\bd x' = f'$. |
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230 This is possible by \nn{...}. |
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231 Finally, define |
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232 \[ |
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233 e(p\ot b) \deq x' \bullet p''(b'') . |
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234 \] |
149 |
235 |
150 |
236 |
151 \medskip |
237 \medskip |
152 |
238 |
153 \nn{to be continued....} |
239 \nn{to be continued....} |