Products in spin$^c$-cobordism

TL;DR

This paper computes the mod 2 spin^c-cobordism ring and its prime spectrum, providing explicit calculations and examples that address longstanding questions in cobordism theory and manifold classification.

Contribution

It offers the first detailed calculation of the mod 2 spin^c-cobordism ring up to inseparable isogeny and constructs a novel subring, advancing understanding of cobordism structures.

Findings

Calculated the mod 2 spin^c-cobordism ring up to inseparable isogeny.

Determined the prime ideal spectrum of the ring.

Provided an explicit example of a 24-dimensional spin manifold not cobordant to a sum of squares.

Abstract

We calculate the mod $2$ spin$^c$-cobordism ring up to uniform $F$-isomorphism (i.e., inseparable isogeny). As a consequence we get the prime ideal spectrum of the mod $2$ spin$^c$-cobordism ring. We also calculate the mod $2$ spin$^c$-cobordism ring ``on the nose'' in degrees $\leq 33$. We construct an infinitely generated nonunital subring of the $2$-torsion in the spin$^c$-cobordism ring. We use our calculations of product structure in the spin and spin$^c$ cobordism rings to give an explicit example, up to cobordism, of a compact $24$-dimensional spin manifold which is not cobordant to a sum of squares, which was asked about in a 1965 question of Milnor.