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Stefan Schwede: Chern classes in equivariant bordism

Time: Tue 2023-05-09 11.00 - 12.00

Location: Cramer room, Albano

Participating: Stefan Schwede (Bonn)

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Complex cobordism MU is arguably the most important cohomology theory in algebraic topology. It represents the bordism theory of stably almost complex manifolds, and it is the universal complex oriented cohomology theory; via Quillen's celebrated theorem, MU is the entry gate for the theory of formal group laws into stable homotopy theory, and thus the cornerstone of chromatic stable homotopy theory.

Tom Dieck's homotopical equivariant bordism MU_G defined with the help of equivariant Thom spaces, strives to be the legitimate equivariant refinement of complex cobordism, for compact Lie groups G. The theory MU_G is the universal equivariantly complex oriented theory; and for abelian compact Lie groups, the coefficient ring MU_G carries the universal G-equivariant formal group law. Homotopical equivariant bordism receives a homomorphism from the geometrically defined equivariant bordism theory; due to the lack of equivariant transversality, this homomorphism is not an isomorphism.

In general, the equivariant bordism ring MU^*_G is still largely mysterious; in this talk, I want to present new information about its structure for unitary groups, and for products of unitary groups. I will introduce Chern classes in U(m)-equivariant homotopical bordism that refine the Conner-Floyd-Chern classes in the MU-cohomology of B U(m). For products of unitary groups, our Chern classes form regular sequences that generate the augmentation ideal of the equivariant bordism rings. Consequently, the Greenlees-May local homology spectral sequence collapses for products of unitary groups. We also use the Chern classes to for a short proof of the MU-completion theorem of Greenlees-May and La Vecchia.