A new construction of lens spaces

TL;DR

This paper introduces a new way to define lens spaces and investigates their classification, showing that all 3-dimensional lens spaces are cobordant to zero in a torus-equivariant manner, with constructive proofs using toric topology.

Contribution

It provides a novel definition of lens spaces and establishes their equivariant cobordism properties through constructive toric topological methods.

Findings

3D lens spaces are $T^2$-equivariantly cobordant to zero.

Certain higher-dimensional lens spaces are $T^{n+1}$-equivariantly cobordant to zero.

Constructive proofs using toric topology techniques.

Abstract

Let $T^n$ be the real $n$-torus group. We give a new definition of lens spaces and study the diffeomorphic classification of lens spaces. We show that any $3$-dimensional lens space $L(p; q)$ is $T^2$-equivariantly cobordant to zero. We also give some sufficient conditions for higher dimensional lens spaces $L(p; q_1, \ldots, q_n)$ to be $T^{n+1}$-equivariantly cobordant to zero. In 2005, B. Hanke showed that complex equivariant cobordism class of a lens space is trivial. Nevertheless, our proofs are constructive using toric topological arguments.