Cellular complexes and embeddings into Euclidean spaces: Möbius strip, torus, and 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.