On the algebraic cobordism ring of involutions

TL;DR

This paper studies the algebraic cobordism ring of involutions on varieties, exploring its structure and applications to fixed locus geometry, including an algebraic analog of Boardman's theorem.

Contribution

It introduces a new framework for the cobordism ring of involutions and establishes an algebraic analog of Boardman's five halves theorem with generalizations.

Findings

Detailed structure of the cobordism ring of involutions

Relations from equivariant K-theory characteristic numbers

Algebraic analog of Boardman's five halves theorem

Abstract

We consider the cobordism ring of involutions of a field of characteristic not two, whose elements are formal differences of classes of smooth projective varieties equipped with an involution, and relations arise from equivariant K-theory characteristic numbers. We investigate in detail the structure of this ring. Concrete applications are provided concerning involutions of varieties, relating the geometry of the ambient variety to that of the fixed locus, in terms of Chern numbers. In particular, we prove an algebraic analog of Boardman's five halves theorem in topology, of which we provide several generalisations and variations.