Timeline for Why the Dold-Thom theorem?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 23, 2014 at 20:20 | comment | added | Chris Gerig | I don't, but I also tend not to gain intuition by making things more abstract (that's a statement about me). Jesse's answer, along with the comments under the question, does seem to provide an intuition about the two things directly; although I will admit that I prefer something similar in spirit to Ryan's suggestion. | |
| Oct 23, 2014 at 19:45 | comment | added | Qiaochu Yuan | Other than that, I don't know what kind of answer you're looking for. An intuition about why two things are isomorphic necessarily involves an intuition about the two things; do you have an intuition for what ko-homology, as opposed to ko-cohomology, is, other than what I wrote down above? (Because I don't, and I'd love one!) | |
| Oct 23, 2014 at 19:10 | comment | added | Qiaochu Yuan | @Chris: you probably won't like this answer, but the way the answer almost has to go is that $F(X)$ is a model for the free $ko$-module spectrum on $X$ (give or take some fuss about basepoints). This is, more or less by definition, the spectrum whose homotopy groups are the $ko$-groups of $X$. | |
| Oct 23, 2014 at 18:47 | comment | added | Chris Gerig | So I think this response shifts my question onto another isomorphism, and my question is then: "Why is $\pi_*(F(X))\cong \widetilde{kO}_*(X)$ true, intuitively?" I see a proof of the isomorphism, but I think it skirts my question. | |
| Oct 13, 2014 at 0:36 | history | edited | Chris Gerig | CC BY-SA 3.0 |
deleted 1 character in body
|
| Oct 9, 2014 at 22:39 | history | edited | მამუკა ჯიბლაძე | CC BY-SA 3.0 |
Had to clarify that kO means connective K-theory
|
| Oct 9, 2014 at 22:38 | comment | added | მამუკა ჯიბლაძე | @ChrisGerig well it is a combination of several propositions whose proofs sort of interleave - (1) that one may (if the basepoint is nice enough) replace $F(X)$ by $Hom(C(X),\mathscr K)$, the latter being the algebra of compact operators on a Hilbert space; (2) that $F(S^n)$ is the $(n-1)$-connected cover of the representing space for $KO^n$; (3) that for sufficiently nice subspaces $Y\subset X$, the map $F(X)\to F(X/Y)$ is a quasifibration with fibre $F(Y)$ (this last step is the analog of the Exercise 3 from Jesse C. McKeown's answer). Btw $kO$ is the connective $K$-theory, I'll correct. | |
| Oct 9, 2014 at 20:38 | comment | added | Chris Gerig | Could you elaborate in your answer about how $\pi_*(F(X))$ relates to reduced K-homology? | |
| Oct 8, 2014 at 7:35 | history | edited | მამუკა ჯიბლაძე | CC BY-SA 3.0 |
deleted 5 characters in body
|
| Oct 8, 2014 at 5:37 | history | edited | მამუკა ჯიბლაძე | CC BY-SA 3.0 |
deleted 9 characters in body
|
| Oct 8, 2014 at 5:32 | history | answered | მამუკა ჯიბლაძე | CC BY-SA 3.0 |