Cobordisms, Manifolds with Torus Action, and Functional Equations

Abstract

The paper is devoted to applications of functional equations to well-known problems of compact torus actions on oriented smooth manifolds. These include the problem of Hirzebruch genera of complex cobordism classes that are determined by complex, almost complex, and stably complex structures on a fixed manifold. We consider actions with connected stabilizer subgroups. For each such action with isolated fixed points, we introduce rigidity functional equations. This is based on the localization theorem for equivariant Hirzebruch genera. We consider actions of maximal tori on homogeneous spaces of compact Lie groups and torus actions on toric and quasitoric manifolds. The arising class of equations contains both classical and new functional equations that play an important role in modern mathematical physics.