Timeline for When are bundles of odd and even differential forms isomorphic?
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 16, 2023 at 8:29 | vote | accept | Ceka | ||
| Sep 21, 2022 at 12:41 | answer | added | Oscar Randal-Williams | timeline score: 3 | |
| Mar 22, 2022 at 21:01 | comment | added | Ceka | If $TM$ is almost complex, setting $V = TM$ gives $$\mathrm{ch}([\Omega^{\text{even}}] - [\Omega^{\text{odd}}]) = \mathbf{e}(TM),$$ which is non-zero if $\chi(M) \neq 0$. Is there an analogue for real K-theory and Stiefel-Whitney classes? (Pretending the above formula still holds, this could give that $\Omega^{\text{even}} \not \simeq \Omega^{\text{odd}}$ if $\chi(M)$ is odd.) | |
| Mar 22, 2022 at 20:53 | comment | added | Ceka | Does this mean that, if $TM$ admits an almost complex structure, then we can relate $[\Omega^{\text{even}}] - [\Omega^{\text{odd}}]$ with the Euler class and solve the problem? In fact, browsing through pages.uoregon.edu/ddugger/kgeom.pdf, Proposition 25.3, I found that $$\mathrm{ch}(e^K(V)) = e^H(V) \cdot \mathrm{Td}^{-1}(V),$$ where $e^K(V) = \sum_i (-1)^i[\Lambda^i V] \in K(M)$ is the K-theory Euler class, $e^H(V)$ is the cohomological Euler class, $\mathrm{ch}$ is the Chern character, and $\mathrm{Td}$ is the Todd class. | |
| Mar 22, 2022 at 15:09 | comment | added | Lennart Meier | After complexification, the difference of $\Omega^{\mathrm{even}}$ and $\Omega^{\mathrm{odd}}$ is the K-theoretic Euler class of $T_{\mathbb{C}}M$. Don't know whether this helps. | |
| Mar 22, 2022 at 9:35 | history | edited | Ceka | CC BY-SA 4.0 |
typo fixed
|
| Mar 21, 2022 at 17:22 | comment | added | LSpice |
TeX points: $\bigoplus_{\text{$k$ even}}$ \bigoplus_{\text{$k$ even}} spaces better, and is more semantically correct, than $\oplus_{k\,\,\mathrm{even}}$ \oplus_{k\,\,\mathrm{even}}. MathJax note: for italics, use stars *stars*, not $fake\ TeX$ $fake\ TeX$. I have edited accordingly.
|
|
| Mar 21, 2022 at 17:21 | history | edited | LSpice | CC BY-SA 4.0 |
TeX and MathJax
|
| Mar 21, 2022 at 17:01 | answer | added | Will Sawin | timeline score: 4 | |
| Mar 21, 2022 at 16:26 | history | edited | Ceka | CC BY-SA 4.0 |
improved formatting
|
| Mar 21, 2022 at 11:40 | history | edited | Ceka | CC BY-SA 4.0 |
improved formatting
|
| Mar 16, 2022 at 13:40 | history | edited | Ceka | CC BY-SA 4.0 |
re-arranged the order of points; added an additional sentence to clarify the $n = 4$ case
|
| Mar 16, 2022 at 12:39 | comment | added | Ceka | That's correct. That $\pi^*\Omega^{\mathrm{odd}} \simeq \pi^*\Omega^{\mathrm{even}}$ gives $\pi^*(c_d(\Omega^{\mathrm{odd}}) - c_d(\Omega^{\mathrm{even}})) = 0$ in $H^{2d}(SM)$, and $\ker \pi^* = \mathbb{Z} \mathbf{e}$ (from Gysin sequence), where $\mathbf{e} \in H^{2d}(M)$ is the Euler class of $TM$, and $\pi^*: H^{2d}(M) \to H^{2d}(SM)$. In the case $\mathbf{e} = 0$ we directly get $\pi_^*$ is injective and $c_d(\Omega^{\mathrm{odd}}) = c_d(\Omega^{\mathrm{even}})$. I prefered to restrict to the case $\chi(M) \neq 0$, as I had dealt with the case $\chi(M) = 0$ in one of the previous points. | |
| Mar 16, 2022 at 11:24 | comment | added | Will Sawin | If $c_d(\Omega^{\textrm{odd}}) - c_d(\Omega^{\textrm{even}})$ is a multiple of $\chi(M)$, doesn't this show equality of the Chern classes (again) if $\chi(M)=0$, not if $\chi(M)\neq 0$? | |
| Mar 16, 2022 at 11:12 | comment | added | Nicolast | Ok ! Thanks for the clarification. | |
| Mar 16, 2022 at 10:27 | history | edited | Ceka | CC BY-SA 4.0 |
corrected a typo in point 5
|
| Mar 16, 2022 at 10:16 | comment | added | Ceka | Thanks for your comment: I corrected the typo in point 3, and added more detail to Point 2. I hope it's clearer now. | |
| Mar 16, 2022 at 10:13 | history | edited | Ceka | CC BY-SA 4.0 |
point 3: typo corrected; point 2: added more detail
|
| Mar 16, 2022 at 7:00 | comment | added | Nicolast | Point 3: you mean that they are isomorphic if and only if M is a torus. Point 2 seems incorrect (you don't even have $\Omega^1(TM) = \Omega^1(V)$. What do you have in mind? | |
| Mar 15, 2022 at 23:29 | history | asked | Ceka | CC BY-SA 4.0 |