Timeline for A generalized Thom Isomorphism
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 26, 2018 at 20:14 | vote | accept | Ali Taghavi | ||
| Mar 6, 2014 at 23:39 | comment | added | Alex Degtyarev | OK, it is not true that you can homotope nonvanishing sections through nonvanishing sections. But it is not needed. By excision, in your notation, $H(E,E_0)=H(D,D_0)$, where $D$ is a "tubular neighborhood" (disk bundle) about the zero section. The same applies to any section; moreover, you can choose these neighborhoods $D_i\supset s_i$ disjoint, so that $H^*$ splits into direct sum. Thus, it remains to discuss one section only (or, rather, one at a time), and then it doesn't matter if it is nonvanishing: just homotope it to zero. | |
| Mar 6, 2014 at 23:32 | comment | added | Ali Taghavi | Thanks, My question was how you apply the excision axiom in the case of my question, that is what is your $B$. Morover could you please answer to last my previous comment? | |
| Mar 6, 2014 at 23:28 | comment | added | Alex Degtyarev | Sorry, I was away. "Excision" means the excision theorem/axiom; the standard wording is $H_*(X,A)=H_*(X\setminus B,A\setminus B)$ if $\operatorname{closure}(B)\subset\operatorname{interior}(A)$. Any section is homotopic to the zero section: $h(x,t)=ts(x)$; ergo, any two sections are homotopic. | |
| Mar 6, 2014 at 23:21 | comment | added | Ali Taghavi | "For one section", is not necessary that we care about the following situation:Assume $s$ and $t$are two non vanishing section.Is there a homotopy of non vanishing sections which connects $s$ to $t$? | |
| Mar 6, 2014 at 23:07 | comment | added | Ali Taghavi | Could you please more explain. What you are considering for excision? | |
| Mar 5, 2014 at 22:28 | history | answered | Alex Degtyarev | CC BY-SA 3.0 |