Timeline for Topological invariance of rational Pontrjagin classes for non-compact spaces
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 6, 2023 at 8:30 | vote | accept | Stefan Kebekus | ||
| Mar 3, 2023 at 16:33 | comment | added | Connor Malin | Let us continue this discussion in chat. | |
| Mar 3, 2023 at 16:27 | comment | added | Igor Belegradek | @ConnorMalin: I should have said it better: how does it imply rational cohomology isomorphism? For simply-connected spaces there is rational Hurewicz. | |
| Mar 3, 2023 at 16:23 | comment | added | Connor Malin | By the LES of a fibration, the map $\mathrm{STOP}/\mathrm{SO} \rightarrow \mathrm{TOP}/\mathrm{O}$ is an equivalence on homotopy groups if $B\mathrm{SO} \rightarrow B\mathrm{O}$ and $B\mathrm{STOP}\rightarrow B\mathrm{TOP}$ are equivalences on homotopy groups. For higher homotopy groups the latter two statements are automatic since this is an inclusion of path components. For the fundamental group, this follows from Kirby's result that $\pi_0(\mathrm{TOP})=\mathbb{Z}/2$. | |
| Mar 3, 2023 at 15:50 | comment | added | Igor Belegradek | @ConnorMalin: sorry, how does it immediately give the nonorientable version? | |
| Mar 3, 2023 at 15:45 | comment | added | Connor Malin | Right, I should have mentioned Kirby proved $\pi_0(\mathrm{TOP})=\mathbb{Z}/2$ which then immediately gives the nonorientable version. | |
| Mar 3, 2023 at 15:40 | comment | added | Igor Belegradek | @ConnorMalin: as far as I can see Proposition 10.2 says that $BSO\to BSTOP$ is a rational homotopy equivalence, which uses that the spaces are simply-connected. The non-orientable case is a bit more hassle. One can reduce to the orientable case by taking Whitney sum of tangent bundles with line bundles corresponding to $w_1$, which is a homeomorphism invariant, and then use the Whitney sum formula for Pontryagin classes to show that $p_i$ is a topological invariant. | |
| Mar 3, 2023 at 15:29 | comment | added | Connor Malin | One place the statement appears in print is Proposition 10.2 of the final Kirby-Siebenmann essay. | |
| Mar 3, 2023 at 15:22 | comment | added | Stefan Kebekus | So this does indeed work for non-compact manifolds! Dear Igor, thank you so much for this extremely helpful reply. I greatly appreciate it! | |
| Mar 3, 2023 at 15:11 | history | answered | Igor Belegradek | CC BY-SA 4.0 |