A non-Hausdorff quaternion multiplication

  • Authors:

  • K. A. Hardie;S. Salbany;J. J. C. Vermeulen;P. J. Witbooi

  • Affiliations:

  • Department of Mathematical Technology, Technikon Pretoria, Private Bag X680, 0001 Pretoria, South Africa;Department of Mathematics, Applied Mathematics and Astronomy, University of South Africa, P.O. Box 392, 0001 Pretoria, South Africa;Department of Mathematics and Applied Mathematics, University of Cape Town, 7700 Rondebosch, South Africa;Department of Mathematics, University of the Western Cape, Private Bag X17, 7535 Bellville, South Africa

  • Venue:
  • Theoretical Computer Science - Topology in computer science
  • Year:
  • 2003

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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.