Stable homotopy at infinity

by Frédéric Déglise (ENS de Lyon)

Thursday 22 Apr 2021, 12:30 → 14:30 Europe/Moscow

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$\mathbb{A}^1$-homotopy theory has open the way to transport topological and differential topology in algebraic geometry. In this long road, many success have been obtained, with substantial reward! In this vein, the question to extend the theory of homotopy groups at infinity has been posed by Aravind Asok in 2008, in a lecture at UCLA. Since then several constructions were in the air.

I will present in this talk a very complete realization of this program in stable $\mathbb{A}^1$-homotopy, that we have recently obtained with Paul Arne Ostvaer and Adrien Dubouloz. While we only produce stable $\mathbb{A}^1$-homotopical invariants, we obtain a theory which directly connects homotopy at infinity with classical cohomological invariants in algebraic geometry such as the interior cohomology, and even vanishing cycles.

Our definition of stable homotopy at infinity is very simple: it is the homotopy cofiber of the natural map from the stable homotopy of a smooth scheme into its compactly supported version. Our work really start in the development of techniques to compute this invariant. A new duality statement allows us to reduce this computation in certain cases to that of the fundamental class of the diagonal (in the open case). The consideration of compactification with boundary a normal corssing divisor lead us to an extension of a formula originally due to Rappoport and Zink which give a presentation of stable homotopy type at infinity with a simple diagram. I will explain how this latter formula allows us to do "Mumford's plumbing" of surfaces (to be recalled in the talk!) in stable $\mathbb{A}^1$-homotopy, with notably a quadratic refinment of intersection matrices.

Notes from the talk Video