Dualizing spheres for compact p-adic analytic groups and duality in chromatic homotopy

Following Boardman-Vogt, McDuff, Segal, and others, we construct a monoidal topological groupoid or space of finite subsets of the plane, and interpret the Burau representation of knot theory as a topological quantum field theory defined on it. Its determinant or writhe is an invertible braided monoidal TQFT which group completes to define a Hopkins-Mahowald model for integral homology as an E 2 E_2 Thom spectrum. We use these ideas to construct an infinite cyclic (Alexander) cover for the space of finite subsets of C \mathbb {C} , and we argue that the TQFT defined by Burau is closely related to the SU(2)-valued Wess-Zumino-Witten model for string theory on R + 3 \mathbb {R}^3_+ .

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The Relative Brauer Group of K(1)-Local Spectra

Mor

2024

International Mathematics Research Notices

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Using profinite Galois descent, we compute the Brauer group of the $K(1)$-local category relative to Morava E-theory. At odd primes this group is generated by a cyclic algebra formed using any primitive $(p-1)$st root of unity, but at the prime two is a group of order $32$ with nontrivial extensions; we give explicit descriptions of the generators, and consider their images in the Brauer group of $KO$. Along the way, we compute the relative Brauer group of completed $KO$, using the étale locally trivial Brauer group of Antieau, Meier and Stojanoska.

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Dualizing spheres for compact p-adic analytic groups and duality in chromatic homotopy

Cited by 3 publications

References 56 publications

Exotic Picard groups and chromatic vanishing via the Gross-Hopkins duality

Exotic Picard groups and chromatic vanishing via the Gross-Hopkins duality

Toward the group completion of the Burau representation

The Relative Brauer Group of K(1)-Local Spectra

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