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Construction and properties of the \(t\)-invariant. (English. Russian original) Zbl 1033.57008

J. Math. Sci., New York 113, No. 6, 849-855 (2003); translation from Zap. Nauchn. Semin. POMI 267, 207-219 (2000).
Special spine theory is a well-known method for representing 3-manifolds. It is known that any compact 3-manifold possesses a special spine and is uniquely recovered from its special spine. Any manifold possesses many different special spines and any two special spines of the same manifold are connected by a sequence of standard moves [see S. V. Matveev, Math. USSR, Izv. 31, 423–434 (1988); translation from Izv. Akad. Nauk SSSR 51, 1104–1116 (1987; Zbl 0676.67002), and R. Piergallini, Topology, 3rd Natl. Meet., Trieste/Italy 1986, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 18, 391–414 (1988; Zbl 0672.57004)].
In the paper under review, special spine theory is used for constructing a new invariant of compact 3-manifolds: the \(t\)-invariant. This is done by assigning to each special polyhedron a number that is preserved under elementary moves. Therefore, the number assigned to a special spine of a 3-manifold \(M\) is a well-defined invariant of \(M\). It is proved that if \(M\) is a connected sum of 3-manifolds \(M_1\) and \(M_2\), then \(t(M)=t(M_1)t(M_2)\), and that one of the Turaev-Viro invariants [see V. G. Turaev and O. Y. Viro, Topology 31, 865–902 (1992; Zbl 0779.57009)] can be expressed in terms of the \(t\)-invariant.
Turaev-Viro invariants fit into the conception of 3-dimensional topological quantum field theory (TQFT), which is a functor from the 3-dimensional bordism category to the category of vector spaces; for Turaev-Viro invariants instead of the bordism category we must consider a category whose objects are surfaces with a fixed 1-dimensional special spine and the morphisms of the category are 3-manifolds with boundary.
In this paper, the \(t\)-invariant admits a similar interpretation. It is also proved that the set of values of the \(t\)-invariant on Seifert manifolds with fixed base and fixed number of singular fibers is finite. A table with the computation of the \(t\)-invariant for all closed irreducible orientable 3-manifolds of complexity at most six completes the paper.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
57R56 Topological quantum field theories (aspects of differential topology)
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