Timeline for Manifolds with polynomial transition maps
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 19, 2017 at 7:26 | comment | added | Ali Taghavi | @js21 Thank you very much for your interesting answer which is an inspiration for this MO queston. | |
| Jul 18, 2017 at 6:10 | comment | added | მამუკა ჯიბლაძე | Oh I see now, sorry. There is an isomorphism of sheaves depending on chosen polynomial atlas. In fact giving such isomorphism should be more or less equivalent to giving a polynomial atlas. Thanks! | |
| Jul 18, 2017 at 5:59 | comment | added | js21 | Ok. I am identifying polynomials functions on a given connected open subset of $\mathbb{R}^n$ with $\mathbb{R}[X_1,\dots,X_n]$. Is it what you wanted to point out ? | |
| Jul 18, 2017 at 5:54 | comment | added | მამუკა ჯიბლაძე | No I don't understand, sorry. Say, $M$ is $\mathbb R$; ${\mathcal F}(M)$ contains a (global) section whose value at $x\in M$ is $x^2$. How to represent this section as a section of the constant sheaf associated to ${\mathbb R}[X]$? Say, one global section of the latter is the one with the same value $X^2$ at any $x$; but that's different, is not it? | |
| Jul 18, 2017 at 5:50 | comment | added | js21 | If $U$ is a polynomial chart, then $\mathcal{F}(U)$ is the set of locally constant functions $U \rightarrow \mathbb{R}[X_1,\dots,X_n]$, i.e. functions on $U$ which are polynomial on each connected component of $U$. This exactly what we want. | |
| Jul 18, 2017 at 5:44 | comment | added | მამუკა ჯიბლაძე | But the sections we need must be all polynomial functions, they are rarely locally constant? | |
| Jul 18, 2017 at 5:43 | comment | added | js21 | A constant sheaf is indeed the sheaf of locally constant functions with values in some given set. A locally constant sheaf (= local system) is a sheaf locally of this form. The latter are in $1$-$1$ correspondance with sets endowed with an action of the fundamental group $\pi$. The action is trivial (e.g. if $\pi = 1$) iff the sheaf is constant. | |
| Jul 17, 2017 at 14:40 | comment | added | მამუკა ჯიბლაძე | What exactly do you mean by a constant sheaf? The ones I know only admit sections corresponding to locally constant functions... | |
| Jul 17, 2017 at 13:12 | history | answered | js21 | CC BY-SA 3.0 |