Timeline for Homology $H_*(TOP, \mathbb{Z}_2)$ of the stable homeomorphism space
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Mar 19, 2017 at 15:00 | comment | added | Neil Strickland | @user51223 I suggest that you convert your comments into a separate answer. | |
| Mar 19, 2017 at 14:26 | comment | added | user51223 | This paper of Madsen also can be helpful projecteuclid.org/download/pdf_1/euclid.pjm/1102868638 | |
| Mar 19, 2017 at 14:21 | comment | added | user51223 | or if it is $\sigma^2$. For $\sigma^2$ it maps nontrivially under the Hurewicz map $h:\pi_{14}G\to H_{14}(G;\mathbb{Z}/2)$ whereas $\kappa$ maps trivially under $h$. I thought this has to help in answering your question. | |
| Mar 19, 2017 at 14:20 | comment | added | user51223 | I comment on the specific question on the kernel of $H_{14}TOP\to H_{14}G$ when restricted to the image of the Hurewicz map $\pi_{14}TOP\to H_{14}TOP$. The space $G/TOP$ at the prime $2$ is a product of EM spaces $K(\mathbb{Z}/2,4n-2)$ and $K(\mathbb{Z}/2,4n)$. So, look at the Serre homotopy exact sequence for $\Omega(G/TOP)\to TOP\to G\to G/TOP\to BTOP\to BG$ which yields $$\pi_{15}G/TOP\to \pi_{14}TOP\to\pi_{14}G\to\pi_{14}G/TOP$$ which reads as $$0\to\pi_{14}TOP\to\mathbb{Z}/2\{\kappa,\sigma^2\}\to\mathbb{Z}/2.$$ Now, I don't now if it is $\kappa$ which lives in $\pi_{14}TOP$ or... | |
| Mar 17, 2017 at 15:48 | comment | added | Denis Nardin | @user106191 $TOP=\Omega BTOP$ so you can probably try to study the Serre spectral sequence. See here for details and references. | |
| Mar 17, 2017 at 8:43 | comment | added | user106191 | Thanks a lot for your answer! I had a look at Madsen-Milgram already, and it seems to me that they compute the homology $H_*(Q_0S^0)$, i.e. $H_*(G)$, and $H_*(G/TOP)$ and $H_*(BTOP)$. However I don't see anything on $H_*(TOP)$, and I can't think of a way to derive it from the other. The other books go in similar directions, mostly focussing on $H_*(G)$ and $H_*(BTOP)$. Do you have an idea on how I could go about $H_*(TOP)$? (I will edit my question, too.) | |
| Mar 16, 2017 at 7:47 | history | answered | Neil Strickland | CC BY-SA 3.0 |