Title: Characteristic Classes and Tate Cohomology.

Abstract:

I will begin by introducing the classical theory of characteristic classes, which associates to each vector bundle over a topological space cohomology classes that measure the obstruction to finding linearly independent sections. I will then describe recent joint work with Kiran Luecke that connects the theory of real and complex characteristic classes to the units in the Tate cohomology rings of the groups C2 and S1, respectively. This perspective offers several conceptual advantages and, in particular, yields a simple proof that the total Chern and Stiefel–Whitney classes arise from maps of spectra—in other words, that they are coherently multiplicative maps of homotopical abelian groups from the K-theories of vector bundles to the multiplicative groups of the corresponding Tate cohomology rings.

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