Complex cobordism with involutions and geometric orientations

We calculate the cobordism ring ${\mathrm{\Omega }}_{\ast }^{C_2}$ of stably almost complex manifolds with involution, and investigate the $C_2$-spectrum ${\mathrm{\Omega }}_{C_2}$ 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 ${\mathrm{\Omega }}_{C_2}$ is the universal example. Examples, in addition to ${\mathrm{\Omega }}_{C_2}$, include the Eilenberg-Maclane spectrum $H \underline{$\mathbb{Z}$}{C_2}$andtheconnectivecover$k{C_2}$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 ${\mathrm{\Omega }}_{C_2}$.