Timeline for Analogy between Stiefel-Whitney and Chern classes
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| May 11, 2020 at 17:42 | comment | added | Lennart Meier | @DenisNardin You're right that it would be actually more natural to use the latter because one doesn't have to use the Thom isomorphism. It was more out of habit. | |
| May 11, 2020 at 16:07 | comment | added | Denis Nardin | Is there a particular reason why you use $M\mathbb{R}$ instead of $\Sigma^\infty BU_{\mathbb{R}}$ or is that only for convenience of computation? | |
| May 11, 2020 at 11:47 | comment | added | Lennart Meier | ...further $M\mathbb{R} \simeq \mathrm{hocolim} \Sigma^{-k-k\sigma} M\mathbb{R}_{k+k\sigma}$ (this is a general fact, see e.g. Section 2 of Hill--Hopkins--Ravenel's Kervaire paper). As $\Phi^{C_2}$ is symmetric monoidal, it also preserves duals and thus takes $\Sigma^{-k-k\sigma}(-) = D(S^{k+k\sigma}) \otimes -$ to $\Sigma^{-k}$ as the $C_2$-fixed points of $S^{k+k\sigma}$ are $S^k$. Thus, we see that the geometric fixed points of $M\mathbb{R}$ are the homotopy colimit over $k$-fold desuspensions of Thom spaces over $BO(n)$, i.e. exactly $MO$. | |
| May 11, 2020 at 11:43 | comment | added | Lennart Meier | Sure. $\Phi = \Phi^{C_2}: \mathrm{Sp}^{C_2} \to \mathrm{Sp}$ is the geometric fixed point functor. It is characterized by the following properties: 1) $\Phi^{C_2}\Sigma^{\infty}X = \Sigma^{\infty}X^{C_2}$; 2) $\Phi^{C_2}$ is symmetric monoidal; 3) $\Phi^{C_2}$ commutes with homotopy colimits. If you want to compute the geometric fixed points of $M\mathbb{R}$ you can argue like follows: $M\mathbb{R}_{k+k\sigma}$ is the Thom space of the universal bundle on $BU(n)$ with the complex conjugation action. Its fixed points are the corresponding Thom space over $BO(n)$... | |
| May 11, 2020 at 11:09 | comment | added | მამუკა ჯიბლაძე | Could you please add few words to mention what $\Phi$ is? Hu and Kriz do explain it referring to Lewis-May-Steinberger, but... | |
| May 11, 2020 at 8:59 | history | answered | Lennart Meier | CC BY-SA 4.0 |