The Linearization Hypothesis

Abstract: Lazard showed that the continuous group cohomology of a large class ofp-adic Lie groups, with p-adic coefficients, satisfies Poincare duality. Analogously to the usual Poincare duality of real manifolds, there are orientability issues, but Lazard showed that the relevant orientation local system is completely determined by the adjoint representation of the group in an explicit manner, allowing for an easy analysis. This can be compared to how the orientation local system on a real manifold is determined by the tangent bundle, a very useful "linearization" of the problem. Now, there is an analogous Poincare duality with spectrum coefficients both in the setting of p-adic Lie groups and in the setting of real manifolds. In the latter case the relevant orientation local system is still determined by the tangent bundle; in fact it is the suspension spectrum of the associated sphere bundle, a statement known as Atiyah duality. In the former case, there is a natural guess for how the orientation local system should still be determined by the adjoint representation. This has been highlighted by recent work of Beaudry-Goerss-Hopkins-Stojanoska in their study of duality for tmf, and they dubbed this guess the "linearization hypothesis". Neither Lazard's techniques nor the usual arguments for Atiyah duality can be used to attack the linearization hypothesis. In this talk I will explain a proof of the linearization hypothesis, whose main ingredients are a deformation of any p-adic Lie group to its Lie algebra, and a rather exotic "cospecialization map" which lets you use this deformation to jump from the Lie algebra to the Lie group as if the deformation were parametrized by a unit interval, even though it is only parametrized by a totally disconnected space.

algebraic topology

Audience: researchers in the topic

MIT topology seminar

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