Timeline for Map which is null-homotopic on compacts
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Apr 6, 2020 at 16:26 | comment | added | Gustavo Granja | The mapping telescope of a sequence of maps is defined in section 3F in Hatcher and a relevant special case is used in the proof of Lemma 2.34 in Hatcher. It's homology groups or homotopy groups are computed by applying the homology or homotopy group functor to the sequence of maps and taking the colimit. Thus when $M$ has the homotopy type of a cell complex, the canonical map from the mapping telescope to $M$ is a homotopy equivalence. | |
| Apr 6, 2020 at 16:17 | answer | added | Gustavo Granja | timeline score: 6 | |
| Apr 6, 2020 at 16:08 | history | edited | erz | CC BY-SA 4.0 |
added a special case to the question
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| Apr 6, 2020 at 16:05 | comment | added | erz | @GustavoGranja could you please elaborate? Especially for somebody who is used to general topology as opposed to the constructions common in algebraic topology. | |
| Apr 6, 2020 at 16:02 | comment | added | erz | @MarkGrant Fubini-Study metric is probably also highly non-compact being а quotient of the Hilbert sphere with respect to $S^1$. | |
| Apr 6, 2020 at 15:19 | comment | added | Gustavo Granja | You can replace $\mathbb CP^\infty$ by the telescope of the inclusions $\mathbb CP^n \hookrightarrow \mathbb CP^{n+1}$ which is locally compact and $\sigma$-compact. | |
| Apr 6, 2020 at 9:46 | comment | added | Mark Grant | There is however a metric on $\mathbb{C}P^\infty$ for which each inclusion $\mathbb{C}P^n\subseteq\mathbb{C}P^\infty$ is an isometric embedding (Fubini-Study). One could ask if the construction of the phantom maps still works with this coarser topology. | |
| Apr 6, 2020 at 9:41 | comment | added | Mark Grant | @erz Ah, OK, it is more subtle than I thought. It appears that $\mathbb{C}P^\infty$ with the weak (CW) topology may not be metrizable, which would imply that it's also not locally compact. I haven't checked, but I assume that the constructions of these phantom maps use the weak topology. | |
| Apr 6, 2020 at 3:29 | comment | added | Wlod AA | A trivial remark: homomorphisms of homotopy groups which are induced by $\ \phi\ $ are trivial. | |
| Apr 5, 2020 at 15:54 | comment | added | erz | @MarkGrant Unfortunately, I am not competent to see if this answers my question. Is $\mathbb{C}P^\infty$ locally compact and $\sigma$-compact? Is every compact subset of it included in a finite skeleton? | |
| Apr 5, 2020 at 11:34 | comment | added | Wlod AA | ANRs are locally contractible. | |
| Apr 5, 2020 at 6:36 | comment | added | Mark Grant | There exist phantom maps $\mathbb{C}P^\infty\to S^3$. These are non-null homotopic maps which become null-homotopic when restricted to each finite skeleton. Doesn't this answer your question in the negative? | |
| Apr 5, 2020 at 6:20 | history | asked | erz | CC BY-SA 4.0 |