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Standard moves for standard polyhedra and spines
No full textA finite 2-dimensional CW-complex P is called a standard (or special) polyhedron if the link of any vertex of P is homeomorphic to a circle with three radii and the link of any other point of its l-skeleton is homeomorphic to a circle with one diameter. Three transformations of standard polyhedra are defined, called standard moves. Moves I and III change a small neighbourhood of a vertex, move II changes a small neighbourhood of an edge. Let P, Q be two standard polyhedra. The following two results are proved: 1) P can be 3-deformed (in the sense of J. H. C. Whitehead) to Q if and only if P can be obtained from Q by a finite sequence of moves I, II, III and its inverses; 2) If P is a spine of a 3-manifold then Q is a spine of the same manifold if and only if P and Q can be related by moves I, II and its inverses only
On four-dimensional 2-handlebodiesand three-manifolds
No full textWe show that for any n > 3 there exists an equivalence functor from the category of n-fold connected simple coverings of B^3 x [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, to the category Chb^{3+1} of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations. As a consequence, we obtain an equivalence theorem for simple coverings of S^3 branched over links, which provides a complete solution to the long-standing Fox-Montesinos covering moves problem. This last result generalizes to coverings of any degree results by the second author and Apostolakis, concerning respectively the case of degree 3 and 4. We also provide an extension of the equivalence theorem to possibly non-simple coverings of S^3 branched over embedded graphs. Then, we factor the functor above through an equivalence functor from H^r to Chb^{3+1}, where H^r is a universal braided category freely generated by a Hopf algebra object H. In this way, we get a complete algebraic description of the category Chb^{3+1}. From this we derive an analogous description of the category Cob^{2+1} of 2-framed relative 3-dimensional cobordisms, which resolves a problem posed by Kerler
On 4-dimensional 2-handlebodies and 3-manifolds
Full text linkWe show that for any n > 3 there exists an equivalence functor from the category of n-fold connected simple coverings of B^3 x [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, to the category Chb^{3+1} of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations. As a consequence, we obtain an equivalence theorem for simple coverings of S^3 branched over links, which provides a complete solution to the long-standing Fox-Montesinos covering moves problem. This last result generalizes to coverings of any degree results by the second author and Apostolakis, concerning respectively the case of degree 3 and 4. We also provide an extension of the equivalence theorem to possibly non-simple coverings of S^3 branched over embedded graphs. Then, we factor the functor above through an equivalence functor from H^r to Chb^{3+1}, where H^r is a universal braided category freely generated by a Hopf algebra object H. In this way, we get a complete algebraic description of the category Chb^{3+1}. From this we derive an analogous description of the category Cob^{2+1} of 2-framed relative 3-dimensional cobordisms, which resolves a problem posed by Kerler
On the generic triangle group and the free metabelian group of rank 2
Full text linkWe introduce the concept of a generic Euclidean triangle and study the group generated by the reflection across the edges of . In particular, we prove that the subgroup of all translations in is free abelian of infinite rank, while the index 2 subgroup of all orientation preserving transformations in is free metabelian of rank 2, with as the commutator subgroup. As a consequence, the group cannot be finitely presented and we provide explicit minimal infinite presentations of both and . This answers in the affirmative the problem of the existence of a minimal presentation for the free metabelian group of rank 2. Moreover, we discuss some examples of non-trivial relations in holding for given non-generic triangles
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