n-dimensional Klein bottles

TL;DR

This paper explores the properties of n-dimensional Klein bottles, including their cohomology, homotopy type, and explicit embeddings, expanding understanding of their topological complexity.

Contribution

It introduces the n-dimensional Klein bottle, analyzing its cohomology, homotopy, and providing explicit embeddings, which were previously unexplored.

Findings

Determined the integral cohomology algebra of n-dimensional Klein bottles.

Established the stable homotopy type of these manifolds.

Provided explicit immersions and embeddings in Euclidean space.

Abstract

An n-dimensional analogue of the Klein bottle arose in our study of topological complexity of planar polygon spaces. We determine its integral cohomology algebra and stable homotopy type, and give an explicit immersion and embedding in Euclidean space.