On Braids and the Noncommutative Galois Theory of Algebra over Stable Homotopy

Jonathan Beardsley (University of Nevada)

Monday 7th February 16:00-17:00 Online + Maths 110


This talk describes two related results in algebraic topology which suggest a deep relationship between the braid groups (and particularly their commutator subgroups) and the stable homotopy groups of spheres. The first result builds on work of Mahowald, Hopkins and others showing that mod-2 and integral homology are equivalent to cobordism theories associated to manifolds whose stable normal bundles classified by maps to BΩ³S³≃Ω²S³ and BΩ³S³≃Ω²S³respectively. Cohen and others have shown that Ω²S³ and the classifying space of the infinite braid group, implying that mod-2 homology is also the cobordism theory for so-called braid oriented manifolds. We will describe joint work with Morava showing that Ω³S³ is essentially the group completion of the classifying space of the infinite writhe-free braid group (i.e. the colimit of the commutator subgroups of the braid groups). Moreover, the fiber sequence Ω²S³→Ω²S³→S¹≃Bℤ manifesting Ω²S³ as the universal cover of Ω²S³ can be obtained from the abelianization exact sequences 1→[Bℤ,Bℤ]→Bℤ→ℤ→1 for the braid groups. This suggests a connection between integral homology and writhe-free braids. The exact nature of this connection remains unclear (although it is further investigated in ongoing work of Morava and Rolfsen).

We will then describe the related fact that the fiber sequence Ω²S³→Ω²S³→S¹, along with a map Ω²S³→BO, makes 𝔽₂ a Hopf-Galois extension of ℤ, and ℤ a Hopf-Galois extension of 𝕊, the sphere spectrum of stable homotopy. Specifically, there is an intermediate Hopf-Galois extension 𝕊→ℤ→𝔽₂ in which the Galois coalgebra of 𝕊→ℤ is Ω²S³, the Galois coalgebra of 𝕊→𝔽₂ is Ω²S³, and the Galois coalgebra of ℤ→𝔽₂ is S¹≃Bℤ. As a result, descent data from derived ℤ-modules (resp. derived 𝔽₂-modules) to 𝕊-modules are equivalent to Ω²S³comodules (resp. Ω²S³ comodules). Given the descriptions of these spaces in terms of classifying spaces of braid groups, Koszul duality would suggest that certain classes of 𝕊-modules may further be described as spectral representations of the braid groups or their commutator subgroups, though this is purely speculative.


The talk will be preceded by a tea time at 3:45pm. The Zoom link for the seminar is https://uofglasgow.zoom.us/j/98078798957 and the passcode is the genus of the two-dimensional sphere (4 letters, all lowercase).

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