Group actions on homology quaternionic projective planes

Weintraub, Steven H.

doi:10.1090/S0002-9939-1978-0515869-8

Group actions on homology quaternionic projective planes
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by Steven H. Weintraub PDF

Proc. Amer. Math. Soc. 70 (1978), 75-82 Request permission

Abstract:

A class of ${{\mathbf {Z}}_p}$-actions, resembling well-known actions on the quaternionic projective plane, is defined and studied. The existence of such actions on a closed homology quaternionic projective plane is shown to imply numerical restrictions on the manifold’s Pontrjagin classes. One consequence is that for $p = 3$, or 5, infinitely many smooth manifolds of this type admit no smooth ${{\mathbf {Z}}_p}$-actions.

References

M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR 236952, DOI 10.2307/1970717
Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
William Browder, Ted Petrie, and C. T. C. Wall, The classification of free actions of cyclic groups of odd order on homotopy spheres, Bull. Amer. Math. Soc. 77 (1971), 455–459. MR 279826, DOI 10.1090/S0002-9904-1971-12736-2
Italo José Dejter, Smooth $S^1$-manifolds in the homotopy type of $CP^3$, Michigan Math. J. 23 (1976), no. 1, 83–95. MR 402789
James Eells Jr. and Nicolaas H. Kuiper, Manifolds which are like projective planes, Inst. Hautes Études Sci. Publ. Math. 14 (1962), 5–46. MR 145544
C. N. Lee, Cyclic group actions on homotopy spheres, Proc. Conf. on Transformation Groups (New Orleans, La., 1967) Springer, New York, 1968, pp. p. 207. MR 0245039
Steven H. Weintraub, Semi-free $Z_{p}$-actions on highly-connected manifolds, Math. Z. 145 (1975), no. 2, 163–185. MR 400268, DOI 10.1007/BF01214781