12:30 pm Tuesday, September 6, 2005
GADGET: The arithmetic of homotopy invariants I: The motivic nature of rational homotopy types. by Martin Olsson (UT Austin) in RLM 9.166
In this series of four lectures, I will discuss various arithmetic aspects of non-abelian invariants of algebraic varieties (e.g. the fundamental group). I intend to make each of the lectures essentially self-contained.
If $\pi $ is a group, its unipotent completion $\pi ^{un}$ is defined to be the universal pro--unipotent group receiving a map from $\pi $. In this lecture I will explain that when $\pi $ is the fundamental group of a pointed variety over a suitable base field (e.g. a number field) the group $\pi ^{un}$ is motivic. In particular, I will discuss Hodge, de Rham, \'etale, and crystalline realizations. I will also explain how this motivic structure leads to restrictions on the homotopy types of algebraic varieties. Submitted by
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