equal
deleted
inserted
replaced
617 \[ |
617 \[ |
618 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
618 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
619 \] |
619 \] |
620 These action maps are required to be associative up to homotopy |
620 These action maps are required to be associative up to homotopy |
621 \nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
621 \nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
622 a diagram like the one in Proposition \ref{CHprop} commutes. |
622 a diagram like the one in Theorem \ref{thm:CH} commutes. |
623 \nn{repeat diagram here?} |
623 \nn{repeat diagram here?} |
624 \nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
624 \nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
625 \end{axiom} |
625 \end{axiom} |
626 |
626 |
627 We should strengthen the above axiom to apply to families of collar maps. |
627 We should strengthen the above axiom to apply to families of collar maps. |
1369 \[ |
1369 \[ |
1370 C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . |
1370 C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . |
1371 \] |
1371 \] |
1372 Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$ |
1372 Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$ |
1373 which fix $\bd M$. |
1373 which fix $\bd M$. |
1374 These action maps are required to be associative up to homotopy, |
1374 These action maps are required to be associative up to homotopy, as in Theorem \ref{thm:CH-associativity}, |
1375 and also compatible with composition (gluing) in the sense that |
1375 and also compatible with composition (gluing) in the sense that |
1376 a diagram like the one in Proposition \ref{CHprop} commutes. |
1376 a diagram like the one in Theorem \ref{thm:CH} commutes. |
1377 \end{module-axiom} |
1377 \end{module-axiom} |
1378 |
1378 |
1379 As with the $n$-category version of the above axiom, we should also have families of collar maps act. |
1379 As with the $n$-category version of the above axiom, we should also have families of collar maps act. |
1380 |
1380 |
1381 \medskip |
1381 \medskip |