A combinatorial characterization of 𝑆³×𝑆¹ among closed 4-manifolds

Cavicchioli, Alberto

doi:10.1090/S0002-9939-1989-0931726-4

A combinatorial characterization of $S^ 3\times S^ 1$ among closed $4$-manifolds
HTML articles powered by AMS MathViewer

by Alberto Cavicchioli

Proc. Amer. Math. Soc. 105 (1989), 1008-1014

DOI: https://doi.org/10.1090/S0002-9939-1989-0931726-4

PDF | Request permission

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

Massimo Ferri and Carlo Gagliardi, The only genus zero $n$-manifold is $S^{n}$, Proc. Amer. Math. Soc. 85 (1982), no. 4, 638–642. MR 660620, DOI 10.1090/S0002-9939-1982-0660620-5
M. Ferri, C. Gagliardi, and L. Grasselli, A graph-theoretical representation of PL-manifolds—a survey on crystallizations, Aequationes Math. 31 (1986), no. 2-3, 121–141. MR 867510, DOI 10.1007/BF02188181
Carlo Gagliardi, Regular imbeddings of edge-coloured graphs, Geom. Dedicata 11 (1981), no. 4, 397–414 (English, with Italian summary). MR 637916, DOI 10.1007/BF00181201
Carlo Gagliardi, Extending the concept of genus to dimension $n$, Proc. Amer. Math. Soc. 81 (1981), no. 3, 473–481. MR 597666, DOI 10.1090/S0002-9939-1981-0597666-0
Carlo Gagliardi, How to deduce the fundamental group of a closed $n$-manifold from a contracted triangulation, J. Combin. Inform. System Sci. 4 (1979), no. 3, 237–252 (English, with Italian summary). MR 586313
Frank Harary, Graph theory, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London 1969. MR 0256911
P. J. Hilton and S. Wylie, Homology theory: An introduction to algebraic topology, Cambridge University Press, New York, 1960. MR 0115161
José María Montesinos, Heegaard diagrams for closed $4$-manifolds, Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977) Academic Press, New York-London, 1979, pp. 219–237. MR 537732
Mario Pezzana, Diagrammi di Heegaard e triangolazione contratta, Boll. Un. Mat. Ital. (4) 12 (1975), no. 3, suppl., 98–105 (Italian, with English summary). Collection in memory of Enrico Bompiani. MR 0425969
Colin Patrick Rourke and Brian Joseph Sanderson, Introduction to piecewise-linear topology, Springer Study Edition, Springer-Verlag, Berlin-New York, 1982. Reprint. MR 665919

Graphs, groups and surfaces