equal
deleted
inserted
replaced
43 Thus we associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic |
43 Thus we associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic |
44 to the standard $k$-ball. |
44 to the standard $k$-ball. |
45 By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
45 By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
46 standard $k$-ball. |
46 standard $k$-ball. |
47 We {\it do not} assume that it is equipped with a |
47 We {\it do not} assume that it is equipped with a |
48 preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. |
48 preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. \nn{List the axiom numbers here, mentioning alternate versions, and also the same in the module section.} |
49 |
49 |
50 Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
50 Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
51 the boundary), we want a corresponding |
51 the boundary), we want a corresponding |
52 bijection of sets $f:\cC(X)\to \cC(Y)$. |
52 bijection of sets $f:\cC(X)\to \cC(Y)$. |
53 (This will imply ``strong duality", among other things.) Putting these together, we have |
53 (This will imply ``strong duality", among other things.) Putting these together, we have |
463 In the case where $\cC(X)$ is the set of all labeled embedded cell complexes $K$ in $X$, |
463 In the case where $\cC(X)$ is the set of all labeled embedded cell complexes $K$ in $X$, |
464 define $\pi^*(K) = \pi\inv(K)$, with each codimension $i$ cell $\pi\inv(c)$ labeled by the |
464 define $\pi^*(K) = \pi\inv(K)$, with each codimension $i$ cell $\pi\inv(c)$ labeled by the |
465 same (traditional) $i$-morphism as the corresponding codimension $i$ cell $c$. |
465 same (traditional) $i$-morphism as the corresponding codimension $i$ cell $c$. |
466 |
466 |
467 |
467 |
468 \addtocounter{axiom}{-1} |
468 %\addtocounter{axiom}{-1} |
469 \begin{axiom}[Product (identity) morphisms] |
469 \begin{axiom}[Product (identity) morphisms] |
470 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
470 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
471 there is a map $\pi^*:\cC(X)\to \cC(E)$. |
471 there is a map $\pi^*:\cC(X)\to \cC(E)$. |
472 These maps must satisfy the following conditions. |
472 These maps must satisfy the following conditions. |
473 \begin{enumerate} |
473 \begin{enumerate} |
590 We call the equivalence relation generated by collar maps and homeomorphisms |
590 We call the equivalence relation generated by collar maps and homeomorphisms |
591 isotopic (rel boundary) to the identity {\it extended isotopy}. |
591 isotopic (rel boundary) to the identity {\it extended isotopy}. |
592 |
592 |
593 The revised axiom is |
593 The revised axiom is |
594 |
594 |
595 \addtocounter{axiom}{-1} |
595 %\addtocounter{axiom}{-1} |
596 \begin{axiom}[\textup{\textbf{[plain version]}} Extended isotopy invariance in dimension $n$.] |
596 \begin{axiom}[\textup{\textbf{[plain version]}} Extended isotopy invariance in dimension $n$.] |
597 \label{axiom:extended-isotopies} |
597 \label{axiom:extended-isotopies} |
598 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
598 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
599 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
599 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
600 Then $f$ acts trivially on $\cC(X)$. |
600 Then $f$ acts trivially on $\cC(X)$. |
608 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
608 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
609 Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
609 Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
610 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
610 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
611 |
611 |
612 |
612 |
613 \addtocounter{axiom}{-1} |
613 %\addtocounter{axiom}{-1} |
614 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
614 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
615 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
615 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
616 \[ |
616 \[ |
617 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
617 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
618 \] |
618 \] |
1432 In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
1432 In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
1433 on $\bd B \setmin N$. |
1433 on $\bd B \setmin N$. |
1434 |
1434 |
1435 For $A_\infty$ modules we require |
1435 For $A_\infty$ modules we require |
1436 |
1436 |
1437 \addtocounter{module-axiom}{-1} |
1437 %\addtocounter{module-axiom}{-1} |
1438 \begin{module-axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act] |
1438 \begin{module-axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act] |
1439 For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes |
1439 For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes |
1440 \[ |
1440 \[ |
1441 C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . |
1441 C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . |
1442 \] |
1442 \] |