Complex cobordism with involutions and geometric orientations

2022/02/02 by Carlisle, Jack
#Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2202.01253

Abstract

We calculate the cobordism ring Ω^C₂_* of stably almost complex manifolds with involution, and investigate the C₂-spectrum Ω_C2 which represents it. We introduce the notion of a geometrically oriented C₂-spectrum, which extends the notion of a complex oriented C₂-spectrum, and of which Ω_C2 is the universal example. Examples, in addition to Ω_C2, include the Eilenberg-Maclane spectrum H \underlineℤ_C2 and the connective cover k_C2 of C₂-equivariant K-theory. On the algebraic side, we define and study filtered C₂-equivariant formal group laws, which are the algebraic structures determined by geometrically oriented C₂-spectra. We prove some of the fundamental properties of filtered C₂-equivariant formal group laws, as well as a universality statement for the filtered C₂-equivariant formal group law determined by Ω_C2.

Related

• A note on the equivariant formal group law of the equivariant complex cobordism ring
• Cobordism of G-manifolds
• Circle-equivariant classifying spaces and the rational equivariant sigma genus
• A note on the equivariant cobordism of generalized Dold manifolds
• Multicurves and equivariant cobordism