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28 November 2017
BOGDAN GHEORGHE (MPIM Bonn/Wayne State University): Motivic Homotopy Theory without Schemes


This is current work in progress with Achim Krause and Nicolas Ricka. Motivic homotopy theory over the complex numbers (or related bases) is becoming a tool for attacking classical problems in algebraic topology, in a similar fashion that equivariant homotopy theory can be. For example, recent work of Isaksen & collaborators extended the calculation of classic stable homotopy groups of spheres this way, and Behrens & collaborators made progress towards the telescope conjecture. One strong relation between classical and motivic homotopy theory over Spec C is the fact that the motivic sphere with the Tate twist inverted is the same data as the classical sphere; and the motivic sphere modulo the Tate twist is describable using algebraic data. It thus becomes plausible to expect a construction of the motivic cellular category without going through motivic spaces, and without even mentioning schemes. In this talk, I will show how we construct a filtered (non-motivic) spectrum resembling the motivic sphere, whose category of modules in filtered spectra recovers the category of (p-completed) cellular motivic spectra over C. If time permits, I will show how we can easily recover known computations, for example by recomputing the motivic Steenrod algebra.

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