Precobordism and cobordism

TL;DR

This paper compares various algebraic cobordism theories, showing equivalences and establishing transformations to operational cobordism, with implications for derived K-theory and virtual pullbacks.

Contribution

It demonstrates the equivalence of a simple universal precobordism theory with a more complex derived algebraic cobordism, and constructs transformations to operational cobordism.

Findings

Universal precobordism agrees with derived algebraic cobordism.

Both theories admit a Grothendieck transformation to operational cobordism.

Strengthens results on operational derived K-theory.

Abstract

The purpose of this article is to compare several versions of bivariant algebraic cobordism constructed previously by the author and others. In particular, we show that a simple construction based on the universal precobordism theory of Annala--Yokura agrees with the more complicated theory of bivariant derived algebraic cobordism constructed earlier by the author, and that both of these theories admit a Grothendieck transformation to operational cobordism constructed by Luis Gonz\'alez--Karu over fields of characteristic 0. The proofs are partly based on convenient universal characterizations of several cobordism theories, which should be of independent interest. Using similar techniques, we also strengthen a result of Vezzosi on operational derived $K$-theory. In the appendix, we give a detailed construction of virtual pullbacks in algebraic bordism, filling the gaps in the construction of Lowrey--Sch\"urg.