A non-Hausdorff quaternion multiplication

Abstract

We denote by (S3)′ the barycentric subdivision of the minimal model S3 of the three-dimensional sphere in the category of finite posets and order-preserving functions, op(X) is the poset obtained by reversing the order relations in a poset X. We describe a finite model of a quaternion multiplication in the form of a morphism op(S3)′×(S3)′→S3 that restricts to weak homotopy equivalences on the axes. For such multiplications a version of Hopf's construction can be defined that yields finite models of non-trivial homotopy classes.