Timeline for What do absolute neighborhood retracts look like?
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 26, 2020 at 6:33 | answer | added | Martin Väth | timeline score: 2 | |
| Jul 1, 2020 at 6:20 | answer | added | Cihan | timeline score: 5 | |
| Jul 3, 2019 at 21:16 | comment | added | Ben Wieland | handles: I believe that if an ANR embeds in the Hilbert cube $Q$, that its product with $Q$ is a $Q$-manifold (ie, homogeneous) and they have a good handle theory. I believe that if such an ANR is homotopy equivalent to a CW complex, then after crossing with the Hilbert cube and also a half-open interval $[0,1)$ they become homemorphic (so the CW cx gives a model of handles). However, for compact ANR there is another obstruction to homeomorphism, the Whitehead torsion. Worse, compact ANR can defy the Wall finiteness obstruction and not have a finite handle structure. | |
| Jul 3, 2019 at 20:53 | answer | added | André Henriques | timeline score: 4 | |
| Jul 1, 2019 at 23:13 | history | edited | Tim Campion | CC BY-SA 4.0 |
added 486 characters in body
|
| Jun 29, 2019 at 20:00 | comment | added | Ben Wieland | Wikipedia says that something very similar to my previous comment works. The Whitehead manifold is the complement in the sphere of a fractal, something like a knotted solenoid, called the Whitehead continuum. If instead of taking the complement, we take the quotient space collapsing the continuum to a point, we get a Bing-like space that becomes a manifold when crossing with $\mathbb R$, but has one bad point. It's probably an ENR. | |
| Jun 29, 2019 at 18:44 | comment | added | Misha | It is not representative (too simple). | |
| Jun 29, 2019 at 18:36 | comment | added | Tim Campion | @Misha Thanks, that's just the kind of thing I'm looking for! Do you have a sense for how "representative" this example is? | |
| Jun 29, 2019 at 18:22 | comment | added | Misha | The simplest example I know is the subset of the plane which is the union of a sequence of segments $s_n$ of length $1/n$ all meeting at their common end-point. It is an ANR but not homeomorphic to a CW complex. | |
| Jun 29, 2019 at 17:23 | history | edited | Tim Campion | CC BY-SA 4.0 |
added 7 characters in body
|
| Jun 29, 2019 at 17:21 | comment | added | Ben Wieland | Bing contracts infinitely many tame arcs. What if we take a single wild arc in $\mathbb R^3$ and contract it to a point? Is the quotient space an ENR but not a CW complex? I think that's what "Product of Euclidean Spaces Modulo an Arc" is about. Probably start with its references. | |
| Jun 29, 2019 at 16:34 | history | edited | Tim Campion |
edited tags
|
|
| Jun 29, 2019 at 16:33 | comment | added | Tim Campion | @BenWieland Point well taken on fractals. And wow, the dogbone space seems complicated! I was hoping there might be a simpler example, especially because being an ANR is a local property (and the dogbone space seems like it was cooked up to have interesting global properties). | |
| Jun 28, 2019 at 20:21 | comment | added | Ben Wieland | A fractal is typically not locally contractible. The Cantor set is not locally contractible. Also, direct from definition: an open neighborhood of the Cantor set in the real line has countably many components, hence has collapsed some. (But the Koch snowflake is a fractal embedding of the circle.) . . . I believe that Bing's Dogbone space is an ANR. But it's very close to a CW complex: its product with $\mathbb R$ is $\mathbb R^4$. | |
| Jun 28, 2019 at 20:06 | history | edited | YCor | CC BY-SA 4.0 |
edited tags, made title understandable
|
| Jun 28, 2019 at 19:54 | history | edited | user64494 | CC BY-SA 4.0 |
Typos are corrected (compare ODE and ODEs).
|
| Jun 28, 2019 at 19:51 | history | asked | Tim Campion | CC BY-SA 4.0 |