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Remarks on the entropy of 3-manifolds
Institution:1. Matematisk Institut, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø Denmark;2. Institut Mittag-Lefer Auravdgen 17, S-182 62 Djursholm, Sweden;3. Raunvisindastofnun Haskolans, University of Iceland, Dunhaga 3, 107 Reykjavik, Iceland;1. Mathematics Institute, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria;2. Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria;1. Departamento de Matemática, Instituto de Matemática e Estatística, Universidade Federal da Bahia, Av. Adhemar de Barros s/n, 40170-110 Salvador, Bahia, Brazil;2. Departamento de Matemática, Universidade Federal de Alagoas, Campus A.S. Simoes s/n, 57072-090 Maceió, Alagoas, Brazil;3. Postdoctoral fellow at Math Section of ICTP, Strada Costiera, 11, I-34151, Trieste, Italy;1. College of Information and Statistics, Guangxi University of Finance and Economics, Nanning, 530003, China;2. College of Business Administration, Guangxi University of Finance and Economics, Nanning, 530003, China;1. Dipartimento di Matematica “Federigo Enriques”, Università degli studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy;2. Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia
Abstract:We give a simple combinatoric proof of an exponential upper bound on the number of distinct 3-manifolds that can be constructed by successively identifying nearest neighbour pairs of triangles in the boundary of a simplicial 3-ball and show that all closed simplicial manifolds that can be constructed in this manner are homeomorphic to S3. We discuss the problem of proving that all 3-dimensional simplicial spheres can be obtained by this construction and give an example of a simplicial 3-ball whose boundary triangles can be identified pairwise such that no triangle is identified with any of its neighbours and the resulting 3-dimensional simplicial complex is a simply connected 3-manifold.
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