Chromatic Fixed Point Theory
Nick Kuhn (University of Virginia)
Abstract: \noindent The study of the action of a finite p-group G on a finite G-CW complex X is one of the oldest topics in algebraic topology. In the late 1930's, P. A. Smith proved that if X is mod p acyclic, then so is XG, its subspace of fixed points. A related theorem of Ed Floyd from the early 1950's says that the dimension of the mod p homology of X will bound the dimension of the mod p homology of XG.
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The study of the Balmer spectrum of the homotopy category of G-spectra has lead to the problem of identifying "chromatic" variants of Smith's theorem, with mod p homology replaced by the Morava K-theories (at the prime p). One such chromatic Smith theorem is proved by Barthel et.al.: if G is a cyclic p-group and X is K(n) acyclic, then XG is K(n−1) acyclic (and this answers questions like this for all abelian p-groups).
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In work with Chris Lloyd, we have been able to show that a chromatic analogue of Floyd's theorem is true whenever a chromatic Smith theorem holds. For example, if G is a cyclic p-group, then the dimension over K(n)∗ of K(n)∗(X) will bound the dimension over K(n−1)∗ of K(n−1)∗(XG).
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The proof that chromatic Smith theorems imply the stronger chromatic Floyd theorems uses the representation theory of the symmetric groups.
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These chromatic Floyd theorems open the door for many applications. We have been able to resolve open questions involving the Balmer spectrum for the extraspecial 2-groups. In a different direction, at the prime 2, we can show quick collapsing of the AHSS computing the Morava K-theory of some real Grassmanians: this is a non-equivariant result.
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In my talk, I'll try to give an overview of some of this.
algebraic topology
Audience: researchers in the topic
Series comments: We meet at varying times on Monday. Click here to join the zoom meeting . The Zoom meeting room number is 132-540-375.
The mailing list for this seminar is the MIT topology google group. Email Mike Hopkins if you want to join the list.
| Organizers: | Jeremy Hahn*, Ishan Levy*, André Lee Dixon*, Haynes Miller* |
| *contact for this listing |