2024/08/27 by Anthony Fraga, Fraga, Anthony
Computer Science · Mathematics · #Digital Image Processing Techniques #Point processes and geometric inequalities #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2408.14882
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 Möbius 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 Möbius 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.