Cellular complexes and embeddings into Euclidean spaces: M\"obius strip, torus, and projective plane

TL;DR

This paper constructs explicit homeomorphisms and their inverses between cellular complexes and Euclidean embeddings of the M"obius strip, torus, and projective plane, providing concrete formulas for these mappings.

Contribution

It provides explicit formulas for homeomorphisms and their inverses between cellular complexes and Euclidean embeddings of key surfaces.

Findings

Explicit homeomorphisms constructed for the M"obius strip, torus, and projective plane.

Inverse formulas for these homeomorphisms are derived.

Embeddings are in 3D for the M"obius strip and torus, and 4D for the projective plane.

Abstract

In algebraic topology, we usually represent surfaces by mean of cellular complexes. This representation is intrinsic, but requires to identify some points through an equivalence relation. On the other hand, embedding a surface in a Euclidean space is not intrinsic but does not require to identify points. In the present paper, we are interested in the M\"obius strip, the torus, and the real projective plane. More precisely, we construct explicit homeomorphisms, as well as their inverses, from cellular complexes to surfaces of 3-dimensional (for the M\"obius strip and the torus) and 4-dimensional (for the projective plane) Euclidean spaces. All the embeddings were already known, but we are not aware if explicit formulas for their inverses exist.