Equivariant Complex Cobordism and Geometric Orientations

Abstract

We calculate the cobordism ring of stably almost complex manifolds with involution, and investigate the equivariant spectrum which represents it. We introduce the notion of geometrically oriented spectra, which extends the notion of complex oriented spectra, and of which the geometric cobordism spectrum is the universal example. Other examples of geometrically oriented spectra include the Eilenberg-Maclane spectrum associated to a constant Mackey functor, and the connective cover of equivariant complex K theory. On the algebraic side, we define and study filtered equivariant formal group laws, which are the algebraic structures determined by geometrically oriented spectra. We prove some of the fundamental properties of filtered equivariant formal group laws, as well as a universality statement for the filtered equivariant formal group law determined by the geometric complex cobordism spectrum.