Euler-Poincare characteristic pair of orientable supermanifolds

TL;DR

This paper extends the concept of Euler-Poincare characteristic to supermanifolds, introducing the Euler-Poincare characteristic pair and exploring transversality in $\Pi$-symmetric supermanifolds.

Contribution

It introduces the Euler-Poincare characteristic pair in supergeometry and investigates transversality in $\Pi$-symmetric supermanifolds, advancing topological invariants in supergeometry.

Findings

Defined the Euler-Poincare characteristic pair for supermanifolds

Analyzed transversality in $\Pi$-symmetric supermanifolds

Extended classical topological invariants to supergeometric context

Abstract

The Euler-Poincare characteristic, or Euler characteristic in short, is a fundamental topological invariant of compact manifolds that plays a crucial role in a variety of geometric and topological situations. From this point of view, we tried to expand on this important concept in supergeometry. In this article, we introduce the Euler-Poincare characteristic pair in the supergeometry. In the final section, we examine transversality in the category of $\Pi$-symmetric supermanifolds.