Motivic homotopy theory (Q1035253)

This paper surveys recent developments in motivic homotopy theory. The goal of this theory is to translate many of the powerful techniques from algebraic topology to the realm of algebraic geometry where it can be applied to problems in algebra and number theory as well. The paper describes how, to each (generalized) cohomology theory, there is an algebraic analogue defined for smooth varieties. For example, the algebraic analogue of singular cohomology is motivic cohomology, and for complex cobordism it is algebraic cobordism. The paper introduces the Morel-Voevodsky unstable homotopy category and defines \(T\)-spectra, which are the algebraic counterparts of spectra (of spaces) used to represent cohomological functors. The last section describes recent results, applications, and open problems in motivic homotopy theory. For example, it discusses the relationship between the Bloch-Kato conjecture, the Beilinson-Lichtenbaum conjecture, and the Quillen--Lichtenbaum conjecture and the construction of algebraic cobordism.