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A combinatorial characterization of $ S\sp 3\times S\sp 1$ among closed $ 4$-manifolds


Author: Alberto Cavicchioli
Journal: Proc. Amer. Math. Soc. 105 (1989), 1008-1014
MSC: Primary 57N13; Secondary 57M15, 57Q05, 57Q15
DOI: https://doi.org/10.1090/S0002-9939-1989-0931726-4
MathSciNet review: 931726
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Abstract: The topological product $ {S^3} \times {S^1}$ is proved to be the unique closed connected $ 4$-manifold of regular genus one. As a consequence, the complex projective plane $ C{P^2}$ has regular genus two.


References [Enhancements On Off] (What's this?)

  • [FG] M. Ferri and C. Gagliardi, The only genus zero $ n$-manifold is $ {S^n}$, Proc. Amer. Math. Soc. 85 (1982), 638-642. MR 660620 (84e:57017)
  • [FGG] M. Ferri, C. Gagliardi and L. Grasselli, A graph-theoretical representation of PL-manifolds. A survey on crystallizations, Aequationes Math. 31 (1986), 121-141. MR 867510 (88a:05057)
  • [G1] C. Gagliardi, Regular imbeddings of edge-coloured graphs, Geom. Dedicata 11 (1981), 397-414. MR 637916 (83g:05032)
  • [G2] -, Extending the concept of genus to dimension $ n$, Proc. Amer. Math. Soc. 81 (1981), 473-481. MR 597666 (82a:57004)
  • [G3] -, How to deduce the fundamental group of a closed $ n$-manifold from a contracted triangulation, J. Combin. Inform. System Sci. 4 (1979), 237-252. MR 586313 (81m:57014)
  • [H] F. Harary, Graph theory, Addison-Wesley, Reading, 1969. MR 0256911 (41:1566)
  • [HW] P. J. Hilton and S. Wylie, An introduction to algebraic topology--Homology theory, Cambridge Univ. Press, Cambridge, 1960. MR 0115161 (22:5963)
  • [M] J. M. Montesinos, Heegaard diagrams for closed $ 4$-manifolds, Geometric Topology (J. Cantrell, ed.), Proc. 1977 Georgia Conference, Academic Press, New York, London, 1979, pp. 219-237. MR 537732 (80k:57022)
  • [P] M. Pezzana, Diagrammi di Heegaard e triangolazione contratta, Boll. Un. Mat. Ital. 12, Suppl. fasc. 3 (1975), 98-105. MR 0425969 (54:13918)
  • [RS] C. Rourke and B. Sanderson, Introduction to piecewise linear topology, Springer-Verlag, Berlin, Heidelberg, New York, 1969. MR 665919 (83g:57009)
  • [W] A. T. White, Graphs, groups and surfaces, North Holland, Amsterdam, 1973.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0931726-4
Keywords: PL-manifold, contracted triangulation, pseudocomplex, multigraph, edge-coloured graph, $ 2$-cell imbedding, crystallization, regular genus
Article copyright: © Copyright 1989 American Mathematical Society

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