Complex cobordism with involutions and geometric orientations

TL;DR

This paper computes the cobordism ring of stably almost complex manifolds with involution, introduces geometrically oriented $C_2$-spectra, and studies associated algebraic structures called filtered $C_2$-equivariant formal group laws.

Contribution

It extends complex orientations to involution-invariant settings and introduces the universal geometrically oriented $C_2$-spectrum, along with algebraic formal group laws.

Findings

Computed the cobordism ring $oldsymbol{ ext{ extOmega}^{C_2}_*}$.

Defined and studied filtered $C_2$-equivariant formal group laws.

Established universality of the formal group law for $oldsymbol{ extOmega_{C_2}}$.

Abstract

We calculate the cobordism ring $\Omega^{C_2}_*$ of stably almost complex manifolds with involution, and investigate the $C_2$-spectrum $\Omega_{C_2}$ which represents it. We introduce the notion of a geometrically oriented $C_2$-spectrum, which extends the notion of a complex oriented $C_2$-spectrum, and of which $\Omega_{C_2}$ is the universal example. Examples, in addition to $\Omega_{C_2}$, include the Eilenberg-Maclane spectrum $H \underline{\mathbb{Z}}_{C_2}$ and the connective cover $k_{C_2}$ of $C_2$-equivariant $K$-theory. On the algebraic side, we define and study filtered $C_2$-equivariant formal group laws, which are the algebraic structures determined by geometrically oriented $C_2$-spectra. We prove some of the fundamental properties of filtered $C_2$-equivariant formal group laws, as well as a universality statement for the filtered $C_2$-equivariant formal group law determined by $\Omega_{C_2}$.