278 \xxpar{Isotopy invariance in dimension $n$ (preliminary version):} |
278 \xxpar{Isotopy invariance in dimension $n$ (preliminary version):} |
279 {Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
279 {Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
280 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
280 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
281 Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$.} |
281 Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$.} |
282 |
282 |
283 We will strengthen the above axiom in two ways. |
283 This axiom needs to be strengthened to force product morphisms to act as the identity. |
284 (Amusingly, these two ways are related to each of the two senses of the term |
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285 ``pseudo-isotopy".) |
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286 |
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287 First, we require that $f$ act trivially on $\cC(X)$ if it is pseudo-isotopic to the identity |
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288 in the sense of homeomorphisms of mapping cylinders. |
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289 This is motivated by TQFT considerations: |
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290 If the mapping cylinder of $f$ is homeomorphic to the mapping cylinder of the identity, |
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291 then these two $n{+}1$-manifolds should induce the same map from $\cC(X)$ to itself. |
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292 \nn{is there a non-TQFT reason to require this?} |
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293 |
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294 Second, we require that product (a.k.a.\ identity) $n$-morphisms act as the identity. |
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295 Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
284 Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
296 Let $J$ be a 1-ball (interval). |
285 Let $J$ be a 1-ball (interval). |
297 We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
286 We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
298 (Here we use the ``pinched" version of $Y\times J$. |
287 (Here we use the ``pinched" version of $Y\times J$. |
299 \nn{need notation for this}) |
288 \nn{need notation for this}) |
304 \end{eqnarray*} |
293 \end{eqnarray*} |
305 (See Figure \ref{glue-collar}.) |
294 (See Figure \ref{glue-collar}.) |
306 \begin{figure}[!ht]\begin{equation*} |
295 \begin{figure}[!ht]\begin{equation*} |
307 \mathfig{.9}{tempkw/glue-collar} |
296 \mathfig{.9}{tempkw/glue-collar} |
308 \end{equation*}\caption{Extended homeomorphism.}\label{glue-collar}\end{figure} |
297 \end{equation*}\caption{Extended homeomorphism.}\label{glue-collar}\end{figure} |
309 We will call $\psi_{Y,J}$ an extended isotopy. |
298 We say that $\psi_{Y,J}$ is {\it extended isotopic} to the identity map. |
310 \nn{or extended homeomorphism? see below.} |
299 \nn{bad terminology; fix it later} |
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300 \nn{also need to make clear that plain old isotopic to the identity implies |
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301 extended isotopic} |
311 \nn{maybe remark that in some examples (e.g.\ ones based on sub cell complexes) |
302 \nn{maybe remark that in some examples (e.g.\ ones based on sub cell complexes) |
312 extended isotopies are also plain isotopies, so |
303 extended isotopies are also plain isotopies, so |
313 no extension necessary} |
304 no extension necessary} |
314 It can be thought of as the action of the inverse of |
305 It can be thought of as the action of the inverse of |
315 a map which projects a collar neighborhood of $Y$ onto $Y$. |
306 a map which projects a collar neighborhood of $Y$ onto $Y$. |
316 (This sort of collapse map is the other sense of ``pseudo-isotopy".) |
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317 \nn{need to check this} |
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318 |
307 |
319 The revised axiom is |
308 The revised axiom is |
320 |
309 |
321 \xxpar{Pseudo and extended isotopy invariance in dimension $n$:} |
310 \xxpar{Extended isotopy invariance in dimension $n$:} |
322 {Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
311 {Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
323 to the identity on $\bd X$ and is pseudo-isotopic or extended isotopic (rel boundary) to the identity. |
312 to the identity on $\bd X$ and is extended isotopic (rel boundary) to the identity. |
324 Then $f$ acts trivially on $\cC(X)$.} |
313 Then $f$ acts trivially on $\cC(X)$.} |
325 |
314 |
326 \nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
315 \nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
327 |
316 |
328 \smallskip |
317 \smallskip |