Groups of PL Λ-homology spheres

Anderson, Gerald A.

doi:10.1090/S0002-9947-1978-0492289-0

Groups of PL $\Lambda$-homology spheres
HTML articles powered by AMS MathViewer

by Gerald A. Anderson PDF

Trans. Amer. Math. Soc. 241 (1978), 55-67 Request permission

Abstract:

Let $\wedge = {{\textbf {Z}}_k}$ be a subring of Q. The group of ${H_K}$-cobordism classes of closed PL n-manifolds with the $\wedge$-homology of ${S^n}$ is computed for $n \ge 4$ (modulo K-torsion). The simply connected version is also computed.

References

J. P. Alexander, G. C. Hamrick, and J. W. Vick, Involutions on homotopy spheres, Invent. Math. 24 (1974), 35–50. MR 345118, DOI 10.1007/BF01418786
J. P. Alexander, G. C. Hamrick, and J. W. Vick, Cobordism of manifolds with odd order normal bundle, Invent. Math. 24 (1974), 83–94. MR 336760, DOI 10.1007/BF01404300
G. A. Anderson, Surgery with coefficients, Lecture Notes in Mathematics, Vol. 591, Springer-Verlag, Berlin-New York, 1977. MR 0451264, DOI 10.1007/BFb0096745

Calculation of the surgery obstruction groups

Jean Barge, Jean Lannes, François Latour, and Pierre Vogel, $\Lambda$-sphères, Ann. Sci. École Norm. Sup. (4) 7 (1974), 463–505 (1975) (French). MR 377939
William Browder, Surgery on simply-connected manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 65, Springer-Verlag, New York-Heidelberg, 1972. MR 0358813, DOI 10.1007/978-3-642-50020-6
Sylvain E. Cappell and Julius L. Shaneson, The codimension two placement problem and homology equivalent manifolds, Ann. of Math. (2) 99 (1974), 277–348. MR 339216, DOI 10.2307/1970901
Heinz Hopf, Fundamentalgruppe und zweite Bettische Gruppe, Comment. Math. Helv. 14 (1942), 257–309 (German). MR 6510, DOI 10.1007/BF02565622
Morris W. Hirsch and Barry Mazur, Smoothings of piecewise linear manifolds, Annals of Mathematics Studies, No. 80, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0415630
Lowell Jones, Patch spaces: a geometric representation for Poincaré spaces, Ann. of Math. (2) 97 (1973), 306–343. MR 315720, DOI 10.2307/1970849
Michel A. Kervaire, Smooth homology spheres and their fundamental groups, Trans. Amer. Math. Soc. 144 (1969), 67–72. MR 253347, DOI 10.1090/S0002-9947-1969-0253347-3
Michel A. Kervaire and John W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR 148075, DOI 10.1090/S0273-0979-2015-01504-1
Robion C. Kirby and Laurence C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Annals of Mathematics Studies, No. 88, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. With notes by John Milnor and Michael Atiyah. MR 0645390, DOI 10.1515/9781400881505
N. Martin, On the difference between homology and piecewise-linear bundles, J. London Math. Soc. (2) 6 (1973), 197–204. MR 314061, DOI 10.1112/jlms/s2-6.2.197
N. Martin and C. R. F. Maunder, Homology cobordism bundles, Topology 10 (1971), 93–110. MR 273634, DOI 10.1016/0040-9383(71)90032-2
Frank Quinn, Semifree group actions and surgery on PL homology manifolds, Geometric topology (Proc. Conf., Park City, Utah, 1974) Lecture Notes in Math., Vol. 438, Springer, Berlin, 1975, pp. 395–414. MR 0402779

A new result in the theory of

dimensional manifolds

8

D. Sullivan, Singularities in spaces, Proceedings of Liverpool Singularities Symposium, II (1969/1970), Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971, pp. 196–206. MR 0339241

Geometric topology: Localization, periodicity and Galois symmetry