158 \xxpar{Product (identity) morphisms:} |
158 \xxpar{Product (identity) morphisms:} |
159 {Let $X$ be homeomorphic to a $k$-ball and $D$ be homeomorphic to an $m$-ball, with $k+m \le n$. |
159 {Let $X$ be homeomorphic to a $k$-ball and $D$ be homeomorphic to an $m$-ball, with $k+m \le n$. |
160 Then we have a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. |
160 Then we have a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. |
161 If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram |
161 If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram |
162 \[ \xymatrix{ |
162 \[ \xymatrix{ |
163 X\times D \ar[r]^{\tilde{f}} \ar[d]^{\pi} & X'\times D' \ar[d]^{\pi} \\ |
163 X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ |
164 X \ar[r]^{f} & X' |
164 X \ar[r]^{f} & X' |
165 } \] |
165 } \] |
166 commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} |
166 commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} |
167 |
167 |
168 \nn{Need to say something about compatibility with gluing (of both $X$ and $D$) above.} |
168 \nn{Need to say something about compatibility with gluing (of both $X$ and $D$) above.} |
174 We start with the plain $n$-category case. |
174 We start with the plain $n$-category case. |
175 |
175 |
176 \xxpar{Isotopy invariance in dimension $n$ (preliminary version):} |
176 \xxpar{Isotopy invariance in dimension $n$ (preliminary version):} |
177 {Let $X$ be homeomorphic to the $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
177 {Let $X$ be homeomorphic to the $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
178 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
178 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
179 Then $f(a) = a$ for all $a\in \cC(X)$.} |
179 Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$.} |
180 |
180 |
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181 We will strengthen the above axiom in two ways. |
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182 (Amusingly, these two ways are related to each of the two senses of the term |
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183 ``pseudo-isotopy".) |
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184 |
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185 First, we require that $f$ act trivially on $\cC(X)$ if it is pseudo-isotopic to the identity |
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186 in the sense of homeomorphisms of mapping cylinders. |
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187 This is motivated by TQFT considerations: |
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188 If the mapping cylinder of $f$ is homeomorphic to the mapping cylinder of the identity, |
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189 then these two $n{+}1$-manifolds should induce the same map from $\cC(X)$ to itself. |
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190 \nn{is there a non-TQFT reason to require this?} |
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191 |
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192 Second, we require that product (a.k.a.\ identity) $n$-morphisms act as the identity. |
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193 Let $X$ be an $n$-ball and $Y\sub\bd X$ be at $n{-}1$-ball. |
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194 Let $J$ be a 1-ball (interval). |
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195 We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
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196 We define a map |
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197 \begin{eqnarray*} |
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198 \psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
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199 a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
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200 \end{eqnarray*} |
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201 \nn{need to say something somewhere about pinched boundary convention for products} |
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202 We will call $\psi_{Y,J}$ an extended isotopy. |
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203 It can be thought of as the action of the inverse of |
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204 a map which projects a collar neighborhood of $Y$ onto $Y$. |
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205 (This sort of collapse map is the other sense of ``pseudo-isotopy".) |
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206 \nn{need to check this} |
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207 |
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208 The revised axiom is |
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209 |
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210 \xxpar{Pseudo and extended isotopy invariance in dimension $n$:} |
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211 {Let $X$ be homeomorphic to the $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
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212 to the identity on $\bd X$ and is pseudo-isotopic or extended isotopic (rel boundary) to the identity. |
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213 Then $f$ acts trivially on $\cC(X)$.} |
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214 |
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215 \nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
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183 |
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219 |
185 \medskip |
220 \medskip |