Landweber exactness of the formal group law in $c_1$-spherical bordism

TL;DR

This paper investigates the algebraic structure of $c_1$-spherical bordism, proving Landweber exactness of its formal group law and establishing conditions for complex orientations after inverting Fermat primes.

Contribution

It demonstrates Landweber exactness for the formal group law in $c_1$-spherical bordism and constructs a complex orientation after inverting Fermat primes.

Findings

Formal group law in $c_1$-spherical bordism is Landweber exact.

Existence of a complex orientation after inverting Fermat primes.

The coefficient ring structure is explicitly described.

Abstract

We describe the structure of the coefficient ring $W^*(pt)=\varOmega_W^*$ of the $c_1$-spherical bordism theory for an arbitrary $SU$-bilinear multiplication. We prove that for any $SU$-bilinear multiplication the formal group of the theory $W^*$ is Landweber exact. Also we show that after inverting the set $\mathcal P$ of Fermat primes there exists a complex orientation of the localized theory $W^*[\mathcal P^{-1}]$ such that the coefficients of the corresponding formal group law generate the whole coefficient ring $\varOmega_W^*[\mathcal P^{-1}]$.