Thread: On o-minimal homotopy groups

We work over an o-minimal expansion of a real closed field. The o-minimal homotopy groups of a definable set are defined naturally using definable continuous maps. We prove that any two semialgebraic maps which are definably homotopic are also semialgebraically homotopic. This result together with known results on semialgebraic homotopy allows us to develop an o-minimal homotopy theory. In particular, we obtain o-minimal versions of the Hurewicz theorems and the Whitehead theorem.



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Sounds interesting.....Although I'm not exactly sure what homotopy groups are defined on things like closed fields....My knowledge of homotopy groups are mainly associated to topological spaces which do have underlying sets associated to them...but these sets typically don't have extra algebraic structure imposed on them as well. I am familiar with Whitehead's theorem though which is about Homotopy groups for topological spaces (Connected Topological Spaces at that I believe)...And so I can only guess that your more general version must have something to do with certain mappings between closed fields that induce isomorphisms for all homotopy groups associated to these clesed fields defining some type of homotopy equivalence in the context of this more general form of homotopy theory? (or something of this nature perhaps?). Anyway it sounds like the kind of thing I could get into if I were to learn a bit more about it. I always liked Algebra as well as topology and the way concepts from one field of mathematics can be generalized to fit the setting of other mathematical fields as well. I will have to read the link to the article you left to see if I can understand the specifics of your theory on something more than just a speculative level.

Freddy