The bordism ring of manifolds equipped with an involution was computed additively by Conner-Floyd (1965) and multiplicatively by Alexander (1972). Alexander’s description is explicit but complicated and doesn’t seem to enjoy a simple algebraic interpretation.
In this talk I will discuss that if one extends the problem and
1) considers the family of bordism rings of manifolds with n commuting involutions for all n, and
2) takes into account the representation sphere-grading,
then there is a simple algebraic universal property. This is joint work with Stefan Schwede.