Complex cobordism with involutions and geometric orientations

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Publication date: 2 February 2022

Abstract: We calculate the cobordism ring

O m e g a_{*}^{C_{2}}

of stably almost complex manifolds with involution, and investigate the

C_{2}

-spectrum

O m e g a_{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

O m e g a_{C_{2}}

is the universal example. Examples, in addition to

O m e g a_{C_{2}}

, include the Eilenberg-Maclane spectrum

H u n d e r l i n e {m a t h b b 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

O m e g a_{C_{2}}

.

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