Speaker: Prof. Andrew Salch, Wayne State University

Title: Derived Picard-Lefschetz theory and K(n)-local stable homotopy groups of finite CW-complexes

Abstract: In classical complex algebraic geometry, given a family of projective complex varieties parameterized over a disk with a single singular point, the local invariant cycle theorem establishes a close connection between the rational cohomology of the singular fiber and the fixed points of the Picard-Lefschetz (i.e., monodromy) operator on the rational cohomology of a smooth fiber. In this talk I will present a derived version of the local invariant cycle theorem which applies to the situation of a family of filtered differential graded algebras parameterized over a disk with a single singular point. Our "derived local invariant cycle theorem" states that the associated spectral sequence for the cohomology of a certain sub-DGA of the singular fiber collapses at its E_2-term. I will explain the main application of this theorem: when used together with earlier results obtained with Mohammad Behzad Kang, we get a proof of an old open conjecture in stable homotopy theory about the cohomology of the unitary group U(n) arising as a subring of the mod p cohomology of the height n Morava stabilizer group. Our proof works by constructing a family of DGAs over a p-adic disk whose smooth fiber is isomorphic to the Chevalley-Eilenberg DGA of the Lie algebra gl_n, and whose singular fiber is the E_1-term of Ravenel's May spectral sequence calculating the mod p cohomology of the height n Morava stabilizer group for large primes p. All results in this talk are joint work with Mohammad Behzad Kang.