175 |
175 |
176 Next we define subcomplexes $G_*^{i,m} \sub CD_*(X)\otimes \bc_*(X)$. |
176 Next we define subcomplexes $G_*^{i,m} \sub CD_*(X)\otimes \bc_*(X)$. |
177 Let $p\ot b$ be a generator of $CD_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b) |
177 Let $p\ot b$ be a generator of $CD_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b) |
178 = \deg(p) + \deg(b)$. |
178 = \deg(p) + \deg(b)$. |
179 $p\ot b$ is (by definition) in $G_*^{i,m}$ if either (a) $\deg(p) = 0$ or (b) |
179 $p\ot b$ is (by definition) in $G_*^{i,m}$ if either (a) $\deg(p) = 0$ or (b) |
180 there exist codimension-zero submanifolds $V_1,\ldots,V_m \sub X$ such that each $V_j$ |
180 there exist codimension-zero submanifolds $V_0,\ldots,V_m \sub X$ such that each $V_j$ |
181 is homeomorphic to a disjoint union of balls and |
181 is homeomorphic to a disjoint union of balls and |
182 \[ |
182 \[ |
183 N_{i,k}(p\ot b) \subeq V_1 \subeq N_{i,k+1}(p\ot b) |
183 N_{i,k}(p\ot b) \subeq V_0 \subeq N_{i,k+1}(p\ot b) |
184 \subeq V_2 \subeq \cdots \subeq V_m \subeq N_{i,k+m}(p\ot b) . |
184 \subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,k+m+1}(p\ot b) . |
185 \] |
185 \] |
186 Further, we require (inductively) that $\bd(p\ot b) \in G_*^{i,m}$. |
186 Further, we require (inductively) that $\bd(p\ot b) \in G_*^{i,m}$. |
187 We also require that $b$ is splitable (transverse) along the boundary of each $V_l$. |
187 We also require that $b$ is splitable (transverse) along the boundary of each $V_l$. |
188 |
188 |
189 Note that $G_*^{i,m+1} \subeq G_*^{i,m}$. |
189 Note that $G_*^{i,m+1} \subeq G_*^{i,m}$. |
190 |
190 |
191 As sketched above and explained in detail below, |
191 As sketched above and explained in detail below, |
192 $G_*^{i,m}$ is a subcomplex where it is easy to define |
192 $G_*^{i,m}$ is a subcomplex where it is easy to define |
193 the evaluation map. |
193 the evaluation map. |
194 The parameter $m$ controls the number of iterated homotopies we are able to construct. |
194 The parameter $m$ controls the number of iterated homotopies we are able to construct |
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195 (Lemma \ref{mhtyLemma}). |
195 The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of |
196 The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of |
196 $CD_*(X)\ot \bc_*(X)$. |
197 $CD_*(X)\ot \bc_*(X)$ (Lemma \ref{xxxlemma}). |
197 |
198 |
198 Next we define a chain map (dependent on some choices) $e: G_*^{i,m} \to \bc_*(X)$. |
199 Next we define a chain map (dependent on some choices) $e: G_*^{i,m} \to \bc_*(X)$. |
199 Let $p\ot b \in G_*^{i,m}$. |
200 Let $p\ot b \in G_*^{i,m}$. |
200 If $\deg(p) = 0$, define |
201 If $\deg(p) = 0$, define |
201 \[ |
202 \[ |
202 e(p\ot b) = p(b) , |
203 e(p\ot b) = p(b) , |
203 \] |
204 \] |
204 where $p(b)$ denotes the obvious action of the diffeomorphism(s) $p$ on the blob diagram $b$. |
205 where $p(b)$ denotes the obvious action of the diffeomorphism(s) $p$ on the blob diagram $b$. |
205 For general $p\ot b$ ($\deg(p) \ge 1$) assume inductively that we have already defined |
206 For general $p\ot b$ ($\deg(p) \ge 1$) assume inductively that we have already defined |
206 $e(p'\ot b')$ when $\deg(p') + \deg(b') < k = \deg(p) + \deg(b)$. |
207 $e(p'\ot b')$ when $\deg(p') + \deg(b') < k = \deg(p) + \deg(b)$. |
207 Choose $V_1$ as above so that |
208 Choose $V = V_0$ as above so that |
208 \[ |
209 \[ |
209 N_{i,k}(p\ot b) \subeq V_1 \subeq N_{i,k+1}(p\ot b) . |
210 N_{i,k}(p\ot b) \subeq V \subeq N_{i,k+1}(p\ot b) . |
210 \] |
211 \] |
211 Let $\bd(p\ot b) = \sum_j p_j\ot b_j$, and let $V_1^j$ be the choice of neighborhood |
212 Let $\bd(p\ot b) = \sum_j p_j\ot b_j$, and let $V^j$ be the choice of neighborhood |
212 of $|p_j|\cup |b_j|$ made at the preceding stage of the induction. |
213 of $|p_j|\cup |b_j|$ made at the preceding stage of the induction. |
213 For all $j$, |
214 For all $j$, |
214 \[ |
215 \[ |
215 V_1^j \subeq N_{i,(k-1)+1}(p_j\ot b_j) \subeq N_{i,k}(p\ot b) \subeq V_1 . |
216 V^j \subeq N_{i,(k-1)+1}(p_j\ot b_j) \subeq N_{i,k}(p\ot b) \subeq V . |
216 \] |
217 \] |
217 (The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.) |
218 (The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.) |
218 We therefore have splittings |
219 We therefore have splittings |
219 \[ |
220 \[ |
220 p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet f'' , |
221 p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet f'' , |
221 \] |
222 \] |
222 where $p' \in CD_*(V_1)$, $p'' \in CD_*(X\setmin V_1)$, |
223 where $p' \in CD_*(V)$, $p'' \in CD_*(X\setmin V)$, |
223 $b' \in \bc_*(V_1)$, $b'' \in \bc_*(X\setmin V_1)$, |
224 $b' \in \bc_*(V)$, $b'' \in \bc_*(X\setmin V)$, |
224 $e' \in \bc_*(p(V_1))$, and $e'' \in \bc_*(p(X\setmin V_1))$. |
225 $e' \in \bc_*(p(V))$, and $e'' \in \bc_*(p(X\setmin V))$. |
225 (Note that since the family of diffeomorphisms $p$ is constant (independent of parameters) |
226 (Note that since the family of diffeomorphisms $p$ is constant (independent of parameters) |
226 near $\bd V_1)$, the expressions $p(V_1) \sub X$ and $p(X\setmin V_1) \sub X$ are |
227 near $\bd V)$, the expressions $p(V) \sub X$ and $p(X\setmin V) \sub X$ are |
227 unambiguous.) |
228 unambiguous.) |
228 We also have that $\deg(b'') = 0 = \deg(p'')$. |
229 We also have that $\deg(b'') = 0 = \deg(p'')$. |
229 Choose $x' \in \bc_*(p(V_1))$ such that $\bd x' = f'$. |
230 Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$. |
230 This is possible by \nn{...}. |
231 This is possible by \nn{...}. |
231 Finally, define |
232 Finally, define |
232 \[ |
233 \[ |
233 e(p\ot b) \deq x' \bullet p''(b'') . |
234 e(p\ot b) \deq x' \bullet p''(b'') . |
234 \] |
235 \] |
235 |
236 |
236 |
237 Note that above we are essentially using the method of acyclic models. |
237 \medskip |
238 For each generator $p\ot b$ we specify the acyclic (in positive degrees) |
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239 target complex $\bc_*(p(V)) \bullet p''(b'')$. |
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240 |
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241 The definition of $e: G_*^{i,m} \to \bc_*(X)$ depends on two sets of choices: |
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242 The choice of neighborhoods $V$ and the choice of inverse boundaries $x'$. |
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243 The next two lemmas show that up to (iterated) homotopy $e$ is independent |
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244 of these choices. |
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245 |
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246 \begin{lemma} |
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247 Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with |
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248 different choices of $x'$ at each step. |
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249 (Same choice of $V$ at each step.) |
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250 Then $e$ and $\tilde{e}$ are homotopic via a homotopy in $\bc_*(p(V)) \bullet p''(b'')$. |
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251 Any two choices of such a first-order homotopy are second-order homotopic, and so on, |
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252 to arbitrary order. |
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253 \end{lemma} |
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254 |
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255 \begin{proof} |
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256 This is a standard result in the method of acyclic models. |
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257 \nn{should we say more here?} |
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258 \nn{maybe this lemma should be subsumed into the next lemma. probably it should.} |
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259 \end{proof} |
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260 |
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261 \begin{lemma} |
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262 Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with |
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263 different choices of $V$ (and hence also different choices of $x'$) at each step. |
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264 If $m \ge 1$ then $e$ and $\tilde{e}$ are homotopic. |
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265 If $m \ge 2$ then any two choices of this first-order homotopy are second-order homotopic. |
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266 And so on. |
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267 In other words, $e : G_*^{i,m} \to \bc_*(X)$ is well-defined up to $m$-th order homotopy. |
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268 \end{lemma} |
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269 |
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270 \begin{proof} |
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271 We construct $h: G_*^{i,m} \to \bc_*(X)$ such that $\bd h + h\bd = e - \tilde{e}$. |
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272 $e$ and $\tilde{e}$ coincide on bidegrees $(0, j)$, so define $h$ |
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273 to be zero there. |
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274 Assume inductively that $h$ has been defined for degrees less than $k$. |
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275 Let $p\ot b$ be a generator of degree $k$. |
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276 Choose $V_1$ as in the definition of $G_*^{i,m}$ so that |
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277 \[ |
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278 N_{i,k+1}(p\ot b) \subeq V_1 \subeq N_{i,k+2}(p\ot b) . |
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279 \] |
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280 There are splittings |
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281 \[ |
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282 p = p'_1\bullet p''_1 , \;\; b = b'_1\bullet b''_1 , |
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283 \;\; e(p\ot b) - \tilde{e}(p\ot b) - h(\bd(p\ot b)) = f'_1\bullet f''_1 , |
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284 \] |
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285 where $p'_1 \in CD_*(V_1)$, $p''_1 \in CD_*(X\setmin V_1)$, |
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286 $b'_1 \in \bc_*(V_1)$, $b''_1 \in \bc_*(X\setmin V_1)$, |
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287 $f'_1 \in \bc_*(p(V_1))$, and $f''_1 \in \bc_*(p(X\setmin V_1))$. |
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288 Inductively, $\bd f'_1 = 0$. |
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289 Choose $x'_1 \in \bc_*(p(V_1))$ so that $\bd x'_1 = f'_1$. |
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290 Define |
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291 \[ |
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292 h(p\ot b) \deq x'_1 \bullet p''_1(b''_1) . |
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293 \] |
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294 This completes the construction of the first-order homotopy when $m \ge 1$. |
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295 |
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296 The $j$-th order homotopy is constructed similarly, with $V_j$ replacing $V_1$ above. |
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297 \end{proof} |
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298 |
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299 Note that on $G_*^{i,m+1} \subeq G_*^{i,m}$, we have defined two maps, |
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300 call them $e_{i,m}$ and $e_{i,m+1}$. |
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301 An easy variation on the above lemma shows that $e_{i,m}$ and $e_{i,m+1}$ are $m$-th |
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302 order homotopic. |
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303 |
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304 |
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305 \medskip |
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306 |
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307 \noop{ |
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308 |
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309 |
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310 \begin{lemma} |
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311 |
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312 \end{lemma} |
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313 \begin{proof} |
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314 |
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315 \end{proof} |
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316 |
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317 |
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318 } |
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319 |
238 |
320 |
239 \nn{to be continued....} |
321 \nn{to be continued....} |
240 |
322 |
241 |
323 |
242 %\nn{say something about associativity here} |
324 %\nn{say something about associativity here} |