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3-manifolds and relativistic kinks. (English) Zbl 0807.57011
Bureš, J. (ed.) et al., The proceedings of the 11th winter school on geometry and physics held in Srní, Czechoslovakia, January 5-12, 1991. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 30, 157-162 (1993).
By \(M\) we mean an orientable, compact, connected 3-manifold without boundary. We say that \(M\) has type 1 if it admits a degree 1 map onto \(\mathbb{R} P^ 3\) (written \(P^ 3)\), otherwise \(M\) is said to have type 2. For example \(P^ 3\) has type 1 and \(S^ 3\) type 2. Type was first studied in [A. R. Shastri, J. G. Williams and author, Int. J. Theor. Phys. 19, 1-23 (1980; Zbl 0448.55009)] in connection with the classification of relativistic kinks, and we explain this connection briefly in §2. We then summarize a few more recent results which help to determine the type of \(M\). These were obtained by A. R. Shastri and the author [Rev. Math. Phys. 3, No. 4, 467-478 (1991; Zbl 0745.57007)], and are illustrated in §3 by determining the types of many 3-manifolds (Table 3.1). The main theorem (2.5) relates the type of \(M\) to \(H_ 1(M;Z)\). Details of the proofs are largely omitted here and can be found in the two cited papers.
For the entire collection see [Zbl 0777.00026].
57N10 Topology of general \(3\)-manifolds (MSC2010)
83C99 General relativity