Timeline for Brouwer fixed point theorem for non-Hausdorff spaces
Current License: CC BY-SA 4.0
22 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 14, 2022 at 8:57 | comment | added | HJRW | @user494312: I said simplicial complexes, which of course are always locally connected. Sorry if I was unclear: I just mean that the FPT for T_0 finite spaces can be proved this way. I have nothing to say about spectral spaces. | |
| Dec 13, 2022 at 19:14 | comment | added | user494312 | @HJRW Don't you need properties like locally connected for good behaviour of singular complexes ? | |
| Dec 13, 2022 at 9:33 | comment | added | user494312 | @Z. M : but actually yes, if you replace 'contractible' and/or 'absolute extensor' by a condition in terms of the maps of finite spaces, you might get a conjecture about fixed property which applies to spectral spaces. The contractability-like condition on $X$ is that $X\to\{pt\}$ lies in the orthogonal wrt the weak lifting property $f^{lr}$ where $f$ is a map of finite topological spaces which is an acyclic Serre fibration. For some $f$ and $X$ a finite CW complex it does define contractability. | |
| Dec 13, 2022 at 9:26 | comment | added | HJRW | @Z.M: thanks for pointing that out, I hadn’t clocked that the T_0 assumption was still in force in the final paragraph. In that case, the relevant fixed point theorem should be a trivial consequence of the Lefschetz FPT for simplicial complexes. (Presumably everyone knows this, but I thought I’d mention it explicitly since no one else has yet.) | |
| Dec 13, 2022 at 9:20 | comment | added | user494312 | @Z. M : As for now, I do not know. I am following up the references by @ Michael Greinecker, and may update the post with references and a conjecture which might apply to spectral spaces. | |
| Dec 13, 2022 at 9:18 | comment | added | user494312 | So is there a form of fixed point theorem for spectral spaces ? | |
| Dec 13, 2022 at 8:35 | comment | added | Z. M | By "this property", I mean the fixed point property that you want — namely, that property on spectral spaces (but that does not incorporate the version for compact Hausdorff spaces that you want). As I explained, spectral spaces are "profinite" sober spaces. | |
| Dec 13, 2022 at 8:31 | comment | added | Z. M | @HJRW OP mentioned this in the post: the necessity of $T_0$. | |
| Dec 12, 2022 at 17:59 | comment | added | HJRW | Without knowing anything about finite topological spaces, I'm surprised that the generalisation of the BFPT you mention is true. I would have guessed that the geometric realisation $|K|$ of the two-point space with the indiscrete topology $K$ is the closed interval. However, the involution that swaps the two points has no fixed point in $K$, but induces a homeomorphism that fixes a point in $|K|$. Apologies if I've misunderstood; I'm entirely ignorant of this subject! | |
| Dec 12, 2022 at 17:10 | comment | added | user494312 | In fact, the modification of contractibility I'd like to use is as follows: X is such that that $X\to \{pt\}$ is in the left-then-right orthogonal of a particular map $M\to\Lambda$ of finite topological spaces which is an acyclic Serre fibration and is a model of barycentric subdivision of the interval, defined in the list here. It is somewhat similar to being an absolute extensor. | |
| Dec 12, 2022 at 17:05 | comment | added | user494312 | Which 'this property'? I think the usual geometric realisation of a poset (viewed as a simplicial set of its chains or some such) is a retract of a filtered colimit of barycentric subdivisions, see this thesis, Theorem 2.4.14 which claims it is at least homotopy equivalent, referenced by P.May. | |
| Dec 12, 2022 at 15:54 | comment | added | Z. M | I mean, the geometric realization of spectral spaces is defined to be the right Kan extension from finite sober spaces, so it will take value in compact Hausdorff spaces. | |
| Dec 12, 2022 at 15:38 | comment | added | Z. M | So does it imply that spectral spaces also have this property, since spectral spaces are characterized by being cofiltered limits of finite sober (or equivalently, $T_0$-)spaces? | |
| Dec 12, 2022 at 13:05 | comment | added | user494312 | @Z.M this is what is defined in the Clader's thesis. ( I seem to recall people writing there was an inaccuracy/error there but do not recall at the moment). Maybe also see this How much of homotopy theory can be done using only finite topological spaces? | |
| S Dec 12, 2022 at 11:38 | history | suggested | Z. M | CC BY-SA 4.0 |
Fix markdown & LaTeX
|
| Dec 12, 2022 at 10:09 | comment | added | Z. M | What is the geometric realization in the last paragraph? | |
| Dec 12, 2022 at 10:08 | review | Suggested edits | |||
| S Dec 12, 2022 at 11:38 | |||||
| Dec 11, 2022 at 21:55 | comment | added | Michael Greinecker | I'm not that knowledgable when it comes to algebraic topology, but "acyclic absolute neighborhood retract" is enough by a fixed-point theorem of Eilenberg and Montgomery. | |
| Dec 11, 2022 at 21:35 | comment | added | user494312 | Thanks! I replaced "contractible" with "absolute extensor for normal spaces", which is different. Would that help ? Is this counterexample an absolute extensor for normal spaces ? | |
| Dec 11, 2022 at 21:29 | history | edited | user494312 | CC BY-SA 4.0 |
replaced "contractible" by "absolute extensor" in response to comment by
|
| Dec 11, 2022 at 19:34 | comment | added | Michael Greinecker | There is a famous example by Kinoshita of a contractible compact subset of $\mathbb{R}^3$ that lacks the topological fixed-point property. | |
| Dec 11, 2022 at 19:26 | history | asked | user494312 | CC BY-SA 4.0 |