A topological interpretation of numbers

Abstract

We survey recent advancements on the realisation problem for mapping degree sets. In particular, we explain that any finite set containing zero is the mapping degree set between some $n$-manifolds, for each $n\ge 3$. Our building factors for the construction of the realising manifolds will be aspherical manifolds. Thus, along the way, we review a couple of open questions related to maps of non-zero degree for aspherical manifolds.

arXiv comment14 pages, survey chapter; v2: final version, to appear in the book "Nisyros 2025: Essays in Geometry, History and Philosophy", edited by A. Papadopoulos and S. Yamada