Galois-equivariant and motivic homotopy

Series: 
Geometry Topology Seminar
Monday, December 9, 2013 - 14:00
1 hour (actually 50 minutes)
Location: 
Skiles 006
,  
MIT
Organizer: 
For a group G, stable G-equivariant homotopy theory studies (the stabilizations of) topological spaces with a G-action up to G-homotopy.  For a field k, stable motivic homotopy theory studies varieties over k up to (a stable notion of) homotopy where the affine line plays the role of the unit interval.  When L/k is a finite Galois extension with Galois group G, there is a functor F from the G-equivariant stable homotopy category to the stable motivic homotopy category of k.  If k is the complex numbers (or any algebraically closed characteristic 0 field) and L=k (so G is trivial), then Marc Levine has shown that F is full and faithful.  If k is the real numbers (or any real closed field) and L=k[i], we show that F is again full and faithful, i.e., that there is a "copy" of stable C_2-equivariant homotopy theory inside of the stable motivic homotopy category of R.  We will explore computational implications of this theorem.This is a report on joint work with Jeremiah Heller.