Torus Manifolds in Equivariant Complex Bordism

TL;DR

This paper characterizes equivariant complex bordism classes of torus manifolds through combinatorial and $K$-theoretic methods, linking geometric structures to algebraic invariants.

Contribution

It provides a complete combinatorial description of equivariant complex bordism for torus manifolds and relates it to equivariant $K$-theory characteristic numbers.

Findings

Oriented torus graphs fully determine equivariant bordism classes.

Equivariant $K$-theory characteristic numbers encode all necessary information.

Omnioriented quasitoric manifolds play a key role in the classification.

Abstract

We restrict geometric tangential equivariant complex $T^n$-bordism to torus manifolds and provide a complete combinatorial description of the appropriate non-commutative ring. We discover, using equivariant $K$-theory characteristic numbers, that the information encoded in the oriented torus graph associated to a stably complex torus manifold completely describes its equivariant bordism class. We also consider the role of omnioriented quasitoric manifolds in this description.