Timeline for Topologies on the infinite join
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 8 at 12:54 | comment | added | Ulrich Pennig | @Tyrone: These are very good examples. Thank you! | |
| Aug 8 at 10:52 | comment | added | Tyrone | To illustrate further: Milnor's classifying space for $S^1$ is metrisable and embeds in the Hilbert cube, whereas the CW topology on the colimit gives you $S^\infty$, which is not metrisable. Both are compactly generated and contractible, and both support continuous free $S^1$ actions. | |
| Aug 8 at 10:45 | comment | added | Tyrone | No. Take for instance the join of a countable discrete space and a single point. One construction gives you a countable wedge of intervals in the CW topology and the other is metrisable and embeds into $\mathbb{R}^2$. Obviously both are compactly generated. | |
| Aug 7 at 6:57 | comment | added | Ulrich Pennig | @Tyrone: So, do the weak and the strong topology on the join agree after taking k-ifications? | |
| Aug 6 at 16:12 | comment | added | Tyrone | Geometric realisation gives a homeomorphism $|X\times Y|\cong |X|\times_k|Y|$, and the compactly generated product $|X|\times_k|Y|$ is $|X|\times|Y|$ if one of $X,Y$ is locally-finite or both $|X|,|Y|$ are countable, but not in general (the full story is marginally more complicated). Things break down quite quickly: the space $|\mathcal{C}_{G,\mathbb{N}}|\times G$ is not compactly generated when $(i)$ $G=\mathbb{R}^\infty$ with the colimit topoloy (this is a countable CW complex), or $(ii)$ $G=\mathbb{R}^\omega$ with the product topology (this is a separable metric space). | |
| Aug 2 at 16:51 | comment | added | Ulrich Pennig | @Tyrone: Ah, yes! That is a good point. Would it not be enough to assume that $G$ is compactly generated? I thought that as long as one works with simplicial compactly generated topological spaces, then geometric realisation preserves finite limits. | |
| Aug 2 at 15:53 | comment | added | Tyrone | The middle mapping in the composition only k-continuous (there is a continuous bijection pointing in the opposite direction). If either $G$ is countable or $|\mathcal{C}_{G,\mathbb{N}}|\times G$ is compactly generated, then the composition is continuous. Otherwise, I don't think its clear. | |
| Aug 1 at 21:28 | history | asked | Ulrich Pennig | CC BY-SA 4.0 |