Timeline for Easy way to define positive higher K groups?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| S Jul 2, 2023 at 6:59 | history | suggested | Jonas Linssen | CC BY-SA 4.0 |
fixed one typo
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| Jul 1, 2023 at 19:34 | review | Suggested edits | |||
| S Jul 2, 2023 at 6:59 | |||||
| Nov 7, 2010 at 11:12 | vote | accept | Ma Ming | ||
| Oct 19, 2010 at 18:45 | comment | added | Johannes Ebert | @Charles Rezk: Maybe that would open an expert's thread, but do you know of written account where the infinite-loop space viewpoint is explicitly combined with the Bott-periodicity? As far as I can see, this amounts to writing down all spaces of the Bott spectrum as $E_\infty$- or $\Gamma$-spaces, plus the Bott maps as maps preserving this structure. | |
| Oct 19, 2010 at 18:40 | comment | added | Johannes Ebert | "though you'll certainly need periodicity to figure out what k ∗ and K ∗ look like" - that is what I mean | |
| Oct 19, 2010 at 18:38 | history | edited | Johannes Ebert | CC BY-SA 2.5 |
added 1657 characters in body
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| Oct 19, 2010 at 18:31 | comment | added | Charles Rezk | I'd disagree with the claim "without Bott periodicity, K-theory is not very interesting". There is a cohomology theory called "connective K-theory" $k^*(X)$, obtained merely using the fact that $Z\times BU$ is an $E_\infty$-space, as in Justin's answer. The groups $k^q(X)$ vanish when $X$ is a space and $q>0$. You can obtain $K^*(X)$ from $k^*(X)$ by inverting the "Bott element" $b\in k^{-2}(*)$, but $K^*$ does not determine $k^*$. You shouldn't need Bott periodicity to carry out these constructions (though you'll certainly need periodicity to figure out what $k^*$ and $K^*$ look like.) | |
| Oct 19, 2010 at 13:07 | comment | added | Ma Ming | Aha! U(n) is homotopy-Abelian in U(2n) (See IOAN JAMES AND EMERY THOMAS "Which Lie Groups are Homotopy-Abelian"), therefore we may say U(\infty) is a homotopy-Abelian group (E_\infty space??). On the other hand, we know that Abelian groups are infinite loop space. Any chance to make this intuition be a concrete proof? | |
| Oct 19, 2010 at 12:32 | comment | added | Ma Ming | Without Bott periodicity, any clue to BU can be delooped infinity times? | |
| Oct 19, 2010 at 8:31 | history | answered | Johannes Ebert | CC BY-SA 2.5 |