Topology of Algebraic Varieties

This is joint work with G .Garkusha. Using the machinery of framed sheaves developed by Voevodsky, a triangulated category of framed motives is introduced and studied. To any smooth algebraic variety $X$, the framed motive $M_{fr}(X)$ is associated in that category. Theorem. The bispectrum \[( M_{fr} X, M_{fr}(X)(1), M_{fr}(X)(2), ... ),\] each term of which is a twisted framed motive of $X$, has motivic homotopy type of the suspension bispectrum of $X$. (this result is an $A^1$-homotopy analog of a theorem due to G.Segal). We also construct a compactly generated triangulated category of framed bispectra and show that it reconstructs the Morel--Voevodsky category $SH(k)$. This machinery allows to recover in characteristic zero the celebrated theorem due to F. Morel stating that the stable $\pi_{0,0}(k)= $ the Grothendiek-Witt ring of the field $k$ . Also this machinery makes approachable Serre finitness conjecture: rational $\pi_{n,0}(\text{the reals}) = 0$ if $n$ is not zero.