  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.
