Timeline for Representing rational homology by manifolds
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 2, 2011 at 21:42 | comment | added | Tom Goodwillie | Yes, of course. Maybe I should have said if $n<k<2n-1$. | |
| Nov 2, 2011 at 18:13 | comment | added | YangMills | Just a small correction, when you say that $\pi_k(S^n)$ is finite if $k>n$, you should exclude the case of $\pi_{4k-1}(S^{2k})$ which is $\mathbb{Z}$ plus finite. | |
| Oct 25, 2011 at 20:00 | comment | added | Lost | As an educational remark, I would like to mention the recent book by Tammo Tom Dieck {Algebraic Topology (EMS Textbooks in Mathematics)} that proves many of the standard theorems of algebraic topology (including Serre's computation of rational homotopy groups) without the use of spectral sequences. I suppose in the question above I am interested in a direct geometric argument that does not secretly involve homotopy groups of spheres. | |
| Oct 25, 2011 at 18:37 | comment | added | Tom Goodwillie | I suppose it must. In the background of all of this stuff, in effect, is the fact that homology groups of spectra are rationally the same as homotopy groups of spectra. I chose to address the rational isomorphism from framed bordism to homology rather than the rational surjection from oriented bordism to homology. This means I could use the five lemma. It also means that I could begin with that fact for the sphere spectrum and deduce it for suspension spectra, but I never needed Thom spectra. | |
| Oct 25, 2011 at 17:45 | comment | added | Lost | I think that Thom's original argument was also based on the computation of homotopy groups of spheres so the argument you sketch is probably not all that different under the surface. Does the proof based on the Atiyah-Hirzebruch spectral sequence use this computation? | |
| Oct 25, 2011 at 17:07 | history | answered | Tom Goodwillie | CC BY-SA 3.0 |