Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the genus of smooth $ 4$-manifolds


Author: Alberto Cavicchioli
Journal: Trans. Amer. Math. Soc. 331 (1992), 203-214
MSC: Primary 57N13; Secondary 57Q05, 57R60
DOI: https://doi.org/10.1090/S0002-9947-1992-1034659-5
MathSciNet review: 1034659
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The projective complex plane and the "twisted" $ {S^3}$ bundle over $ {S^1}$ are proved to be the unique closed prime connected (smooth or PL) $ 4$-manifolds of genus two. Then the classification of the nonorientable $ 4$-manifolds of genus $ 4$ is given. Finally the genus of a manifold $ M$ is shown to be related with the $ 2$nd Betti number of $ M$ and some applications are proved in the general (resp. simply-connected) case.


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

  • [1] J. Bracho and L. Montejano, The combinatorics of colored triangulations of manifolds, Geom. Dedicata 22 (1987), 303-328. MR 887580 (88k:57028)
  • [2] A. Cavicchioli, A combinatorial characterization of $ {S^3} \times {S^1}$ among closed $ 4$-manifolds, Proc. Amer. Math. Soc. 105 (1989), 1008-1014. MR 931726 (89h:57011)
  • [3] S. K. Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983), 279-315. MR 710056 (85c:57015)
  • [4] -, Connections, cohomology and the intersection forms of $ 4$-manifolds, J. Differential Geom. 24 (1986), 275-341. MR 868974 (88g:57033)
  • [5] M. Ferri, Una rappresentazione delle $ n$-varietà topologiche triangolabili mediante grafi $ (n + 1)$-colorati, Boll. Un. Mat. Ital. 13-B (1976), 250-260. MR 0428336 (55:1361)
  • [6] 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)
  • [7] -, On the genus of $ 4$-dimensional products of manifolds, Geom. Dedicata 13 (1982), 331-345. MR 690678 (84m:57010)
  • [8] 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)
  • [9] M. H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), 357-453. MR 679066 (84b:57006)
  • [10] C. Gagliardi, Extending the concept of genus to dimension $ n$, Proc. Amer. Math. Soc. 81 (1981), 473-481. MR 597666 (82a:57004)
  • [11] -, How to deduce the fundamental group of a closed $ n$-manifold from a contracted triangulation, J. Combin. Informat. Systems Sci. 4 (1979), 237-252. MR 586313 (81m:57014)
  • [12] -, On the genus of $ C{P^2}$, Aequationes Math. 37 (1989), 130-140. MR 1004490 (90f:57026)
  • [13] C. Gagliardi and G. Volzone, Handles in graphs and sphere bundles over $ {S^1}$, European J. Combin. 8 (1987), 151-158. MR 896128 (88k:57017)
  • [14] P. J. Hilton and S. Wylie, An introduction to algebraic topology--Homology theory, Cambridge Univ. Press, Cambridge, 1960. MR 0115161 (22:5963)
  • [15] R. Mandelbaum, Four-dimensional topology: an introduction, Bull. Amer. Math. Soc. (N.S.) 2 (1980), 1-159. MR 551752 (81j:57001)
  • [16] J. M. Montesinos, Heegaard diagrams for closed $ 4$-manifolds, Geometric Topology (J. Cantrell, Ed.), Proc. 1977 Georgia Conference, Academic Press, New York, 1979, pp. 219-237. MR 537732 (80k:57022)
  • [17] M. Pezzana, Sulla struttura topologica delle variet'a compatte, Atti Sem. Mat. Fis. Univ. Modena 23 (1974), 269-277. MR 0402792 (53:6606)
  • [18] D. Rolfsen, Knots and links, Math. Lecture Ser. 7, Publish or Perish, Berkeley, Calif., 1976. MR 0515288 (58:24236)
  • [19] A. T. White, Graphs, groups and surface, North-Holland, Amsterdam, 1973.
  • [20] J. H. C. Whitehead, Manifolds with transverse fields in euclidean space, Ann. of Math. (2) 73 (1961), 154-211. MR 0124917 (23:A2225)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57N13, 57Q05, 57R60

Retrieve articles in all journals with MSC: 57N13, 57Q05, 57R60


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1034659-5
Keywords: Smooth $ 4$-manifold, crystallization, genus, collapsing, intersection form, homotopy sphere
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society