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Greg Arone: On the non-commutative stable homotopy category

Time: Tue 2019-04-23 13.15 - 15.00

Location: Room 3418, Lindstedtsvägen 25, 4th floor, Department of Mathematics KTH

Participating: Greg Arone (SU)

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ABSTRACT: A classic theorem of Gelfand-Naimark says there is an equivalence of categories between compact Hausdorff spaces and the opposite of commutative C^*-algebras. This correspondence led to the point of view that C^*-algebras are non-commutative generalizations of topological spaces. The study of C^*-algebras from this perspective is the subject of non-commutative geometry, topology and homotopy theory. I will talk about the category of “non-commutative spectra”, which is the opposite of the stable homotopy category of C^*-algebras.

In the first half we will introduce a stable infinity-category NSp, which we call the category of non-commutative CW spectra. Roughly speaking, NSp is the opposite of the stabilization of the infinity-category of C^* algebras. In the second half we will analyze the structure of NSp. To do this, we introduce a natural filtration of NSp, called the rank filtration. It is analogous to the rank filtration of K-theory. We will describe explicitly the subquotients of the rank filtration of NSp. The description involves certain complexes L_m, which also appear in the description of the quotients of the rank filtration of topological complex K-theory. As a consequence, we can give a very explicit model for the rationalization of NSp. (joint with Ilan Barnea and Tomer Schlank)