Timeline for Simply connected finite CW-complex with only finitely many nontrivial homotopy and homology groups
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| May 8, 2019 at 22:07 | history | edited | YCor | CC BY-SA 4.0 |
added missing hypothesis
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| Jun 21, 2015 at 19:03 | comment | added | Jens Reinhold | Thnaks for this comment. But in fact I was thinking about ${finite}$ CW-complexes, so David's answers was what I was looking for. | |
| Jun 20, 2015 at 18:27 | comment | added | Ricardo Andrade | Alternatively, fix an odd natural number $n$, and consider the Eilenberg–MacLane space $K(\mathbb{Q},n)$; it has the same homotopy type as $X_n$ above. One can prove that $K(\mathbb{Q},n)$ has reduced homology concentrated in degree $n$ by applying inductively the Serre spectral sequence for the fibre sequences $K(\mathbb{Q},i-1) \to \ast \to K(\mathbb{Q},i)$. | |
| Jun 20, 2015 at 18:11 | comment | added | Ricardo Andrade | Consider the rationalization of the $n$-sphere, call it $X_n$, defined as the mapping telescope of a sequence of maps $S^n \to S^n \to S^n \to \cdots$ in which the $i$-th map has degree $i$. The space $X_n$ has just one non-trivial reduced homology group. Moreover, the homotopy groups of $X_n$ can also be computed knowing that sequential homotopy colimits commute with homotopy groups. It follows that $X_n$ has non-trivial homotopy groups in at most two degrees, $n$ and perhaps $2n-1$, which correspond precisely to the homotopy groups of the $n$-sphere which are infinite. | |
| May 24, 2015 at 18:43 | vote | accept | Jens Reinhold | ||
| May 24, 2015 at 18:43 | vote | accept | Jens Reinhold | ||
| May 24, 2015 at 18:43 | |||||
| May 24, 2015 at 7:33 | answer | added | David C | timeline score: 28 | |
| May 24, 2015 at 7:13 | history | asked | Jens Reinhold | CC BY-SA 3.0 |