Circle actions on homotopy spheres not bounding spin manifolds
HTML articles powered by AMS MathViewer
- by Reinhard Schultz
- Trans. Amer. Math. Soc. 213 (1975), 89-98
- DOI: https://doi.org/10.1090/S0002-9947-1975-0380853-6
- PDF | Request permission
Abstract:
Smooth circle actions are constructed on odd-dimensional homotopy spheres that do not bound spin manifolds. Examples are given in every dimension for which exotic spheres of the described type exist.References
- J. F. Adams, Vector fields on spheres, Ann. of Math. (2) 75 (1962), 603–632. MR 139178, DOI 10.2307/1970213
- J. F. Adams, On the groups $J(X)$. I, Topology 2 (1963), 181–195. MR 159336, DOI 10.1016/0040-9383(63)90001-6
- J. F. Adams, On the groups $J(X)$. IV, Topology 5 (1966), 21–71. MR 198470, DOI 10.1016/0040-9383(66)90004-8
- J. F. Adams and G. Walker, On complex Stiefel manifolds, Proc. Cambridge Philos. Soc. 61 (1965), 81–103. MR 171285, DOI 10.1017/S0305004100038688
- D. W. Anderson, E. H. Brown Jr., and F. P. Peterson, The structure of the Spin cobordism ring, Ann. of Math. (2) 86 (1967), 271–298. MR 219077, DOI 10.2307/1970690
- M. F. Atiyah, Thom complexes, Proc. London Math. Soc. (3) 11 (1961), 291–310. MR 131880, DOI 10.1112/plms/s3-11.1.291
- M. F. Atiyah, $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1967. Lecture notes by D. W. Anderson. MR 0224083
- Raoul Bott, The stable homotopy of the classical groups, Ann. of Math. (2) 70 (1959), 313–337. MR 110104, DOI 10.2307/1970106
- Glen E. Bredon, A $\Pi _\ast$-module structure for $\Theta _\ast$ and applications to transformation groups, Ann. of Math. (2) 86 (1967), 434–448. MR 221518, DOI 10.2307/1970609
- William Browder, Surgery and the theory of differentiable transformation groups, Proc. Conf. on Transformation Groups (New Orleans, La., 1967) Springer, New York, 1968, pp. 1–46. MR 0261629
- 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
- G. Brumfiel, On the homotopy groups of $\textrm {BPL}$ and $\textrm {PL/O}$, Ann. of Math. (2) 88 (1968), 291–311. MR 234458, DOI 10.2307/1970576 G. Brumfiel, Differentiable ${S^1}$ actions on homotopy spheres, Univ. of Calif., Berkeley, 1968 (mimeographed).
- Robert R. Clough, The $Z_{2}$ cohomology of a candidate for $B_{Im(J)}$, Illinois J. Math. 14 (1970), 424–433. MR 263069
- H. Blaine Lawson Jr. and Shing Tung Yau, Scalar curvature, non-abelian group actions, and the degree of symmetry of exotic spheres, Comment. Math. Helv. 49 (1974), 232–244. MR 358841, DOI 10.1007/BF02566731
- J. Levine, Semi-free circle actions on spheres, Invent. Math. 22 (1973), 161–186. MR 350767, DOI 10.1007/BF01392300 J. P. May, On kO-oriented bundle theories, Univ. of Chicago, 1974 (mimeographed).
- John W. Milnor, Remarks concerning spin manifolds, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 55–62. MR 0180978
- Daniel Quillen, The Adams conjecture, Topology 10 (1971), 67–80. MR 279804, DOI 10.1016/0040-9383(71)90018-8
- Reinhard E. Schultz, Smooth structures on $S^{p}\times S^{q}$, Ann. of Math. (2) 90 (1969), 187–198. MR 250321, DOI 10.2307/1970687
- Reinhard Schultz, The nonexistence of free $S^{1}$ actions on some homotopy spheres, Proc. Amer. Math. Soc. 27 (1971), 595–597. MR 271985, DOI 10.1090/S0002-9939-1971-0271985-3 —, Differentiable group actions on homotopy spheres. I, II (to appear).
- James D. Stasheff, The image of $J$ as a space $\textrm {mod}$ $p$, Conf. on Algebraic Topology (Univ. of Illinois at Chicago Circle, Chicago, Ill., 1968) Univ. of Illinois at Chicago Circle, Chicago, Ill., 1969, pp. 276–287. MR 0258032 —(Editor), Problems proposed at the Conference, Univ. of Illinois at Chicago, Chicago, Ill., 1969, pp. 288-293.
- C. T. C. Wall, Surgery on compact manifolds, London Mathematical Society Monographs, No. 1, Academic Press, London-New York, 1970. MR 0431216
Bibliographic Information
- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 213 (1975), 89-98
- MSC: Primary 57E25
- DOI: https://doi.org/10.1090/S0002-9947-1975-0380853-6
- MathSciNet review: 0380853