Chern classes in precobordism theories

TL;DR

This paper develops Chern classes within the universal precobordism framework, linking it to Grothendieck rings and Chow rings, and extends projective bundle formulas to broader contexts.

Contribution

It constructs Chern classes in universal precobordism over arbitrary Noetherian bases and relates these to classical invariants like Grothendieck and Chow rings.

Findings

Grothendieck ring recovered from universal precobordism ring

Constructed candidates for Chow rings satisfying Grothendieck--Riemann--Roch

Extended weak projective bundle formula to all projective bundles

Abstract

We construct Chern classes of vector bundles in the universal precobordism theory of Annala--Yokura over an arbitrary Noetherian base ring of finite Krull dimension. As an immediate corollary of this, we show that the Grothendieck ring of vector bundles can be recovered from the universal precobordism ring, and that we can construct candidates for Chow rings satisfying an analogue of the classical Grothendieck--Riemann--Roch theorem. We also strengthen the weak projective bundle formula of Annala--Yokura to work for arbitrary projective bundles.