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)

 
 

 

Branched coverings. I


Author: R. E. Stong
Journal: Trans. Amer. Math. Soc. 276 (1983), 375-402
MSC: Primary 57M12; Secondary 57N70
DOI: https://doi.org/10.1090/S0002-9947-1983-0684516-3
MathSciNet review: 684516
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper analyzes the possible cobordism classes $ [M] - (\deg \;\phi )[N]$ for $ \phi : M \to N$ a smooth branched covering of closed smooth manifolds. It is assumed that the branch set is a codimension $ 2$ submanifold. The results are a fairly complete description in the unoriented case, a partial description in the oriented case, and a detailed analysis of the case in which $ N$ is a sphere.


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

  • [1] M. F. Atiyah, Bordism and cobordism, Proc. Cambridge Philos. Soc. 57 (1961), 200-208. MR 0126856 (23:A4150)
  • [2] I. Berstein and A. L. Edmonds, The degree and branch set of a branched covering, Invent. Math. 45 (1978), 213-220. MR 0474261 (57:13908)
  • [3] N. Brand, Necessary conditions for the existence of branched coverings, Invent. Math. 54 (1979), 1-10. MR 549541 (81b:57001)
  • [4] -, Classifying spaces for branched coverings, Indiana Univ. Math. J. 29 (1980), 229-248. MR 563208 (81c:57001)
  • [5] J. Brillhart, J. Tonascia and P. Weinberger, On the Fermat quotient, Computers in Number Theory (A. O. L. Atkin and B. J. Birch, eds.), Academic Press, New York, 1971, pp. 213-222. MR 0314736 (47:3288)
  • [6] P. E. Conner and E. E. Floyd, Differentiable periodic maps, Springer-Verlag, Berlin, 1964. MR 0176478 (31:750)
  • [7] A. L. Edmonds, Orientability of fixed point sets, Proc. Amer. Math. Soc. 82 (1981), 120-124. MR 603614 (82h:57032)
  • [8] A. Hattori, Genera of ramified coverings, Math. Ann. 195 (1972), 208-226. MR 0290399 (44:7580)
  • [9] F. Hirzebruch, The signature of ramified coverings, Global Analysis, Papers in honor of K. Kodaira, Univ. of Tokyo Press, Tokyo and Princeton Univ. Press, Princeton, N. J., 1969, pp. 252-265. MR 0258060 (41:2707)
  • [10] N. H. Kuiper, The quotient space of $ {\mathbf{CP}}(2)$ by complex conjugation is the $ 4$-sphere, Math. Ann. 208 (1974), 175-177. MR 0346817 (49:11541)
  • [11] P. S. Landweber, Fixed point free conjugations on complex manifolds, Ann. of Math. (2) 86 (1967), 491-502. MR 0220317 (36:3382)
  • [12] C. N. Lee and A. Wasserman, Equivariant characteristic numbers, Proc. 2nd Conf. Compact Transformation Groups (Univ. of Mass., Amherst, 1971), Lecture Notes in Math., vol. 298, Springer-Verlag, Berlin and New York, 1972, pp. 191-216. MR 0365601 (51:1853)
  • [13] M. Nakaoka, Homology of the infinite symmetric group, Ann. of Math. 73 (1961), 229-257. MR 0131874 (24:A1721)
  • [14] -, Note on cohomology algebras of symmetric groups, J. Math. Osaka City Univ. 13 (1962), 45-55. MR 0154905 (27:4849)
  • [15] H. L. Rosenzweig, Bordism of involutions on manifolds, Illinois J. Math. 16 (1972), 1-10. MR 0290386 (44:7568)
  • [16] R. E. Stong, Complex and oriented equivariant bordism, Topology of Manifolds (J. C. Cantrell and C. H. Edwards, Jr., eds.), Markham, Chicago, Ill., 1970, pp. 291-316. MR 0273644 (42:8521)
  • [17] -, On fibering of cobordism classes, Trans. Amer. Math. Soc. 178 (1973), 431-447. MR 0315733 (47:4282)
  • [18] C. T. C. Wall, Determination of the cobordism ring, Ann. of Math. 72 (1960), 292-311. MR 0120654 (22:11403)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57M12, 57N70

Retrieve articles in all journals with MSC: 57M12, 57N70


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0684516-3
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society