Complex cobordism with involutions and geometric orientations

Abstract

We calculate the cobordism ring ${\Omega }_{\ast }^{C_2}$ 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 ${\Omega }_{C_2}$.

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