Timeline for Can a nontrivial spectrum smash to zero with $K$-theory?
Current License: CC BY-SA 3.0
4 events
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| Jun 10, 2012 at 15:32 | comment | added | Sean Tilson | @Neil: I know that argument works more generally. Would you mind posting it as a separate answer? | |
| Jun 10, 2012 at 15:01 | comment | added | Tom Goodwillie | Neil, that's nice. The proof that I know consists of examining the effect of the Bott map $S^2\wedge BU\to BU$ on integral homology. This comes down to calculating the Chern class(es) of the generator of $\tilde K(S^{2n})$, which can be done using the Chern character. | |
| Jun 10, 2012 at 11:36 | comment | added | Neil Strickland |
Moreover, you can prove this without too much computation. Both $K$ and $H/p$ are complex oriented, so a quite conceptual argument shows that the additive formal group law (associated to $H/p$) and the multiplicative one (associated to $K$) become isomorphic over $\pi_*(K\wedge H/p)$. A simple algebraic argument now shows that $\pi_*(K\wedge H/p)=0$.
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| Jun 10, 2012 at 1:59 | history | answered | Tom Goodwillie | CC BY-SA 3.0 |