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On the ARF Invariant of an Involution

Published online by Cambridge University Press:  20 November 2018

Peter Orlik
Peter Orlik*
Affiliation:
Institute of Advanced Study, Princeton, New Jersey; University of Wisconsin, Madison, Wisconsin
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Let Σ4k+1 denote a smooth manifold homeomorphic to the (4k + 1)-sphere, S4k+1k ≧ 1, and T: Σ4k+1 → Σ4k+1 a differentiate free involution. Our aim in this note is to derive a connection between the differentiate structure on Σ4k+1 and the properties of the free involution T.

To be more specific, recall [5] that the h-cobordism classes of smooth manifolds homeomorphic (or, what is the same, homotopy equivalent) to S4k+1, k ≧ 1, form a finite abelian group θ4k+1 with group operation connected sum. The elements are called homotopy spheres. Those homotopy spheres that bound parallelizable manifolds form a subg roup bP4k+2 ⊂ θ4k+1. It is proved in [5, Theorem 8.5] that bP4k+2 is either zero or cyclic of order 2. In the latter case the two distinct homotopy spheres are distinguished by the Arf invariant of the parallelizable manifolds they bound.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 3 , 01 June 1970 , pp. 519 - 524
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Berstein, I., Involutions with nonzero Arf invariant, Bull. Amer. Math. Soc. 74 (1968), 678–682.Google Scholar
2. Browder, W. and Livesay, G. R., Fixed point free involutions on homotopy spheres, Bull. Amer. Math. Soc. 73 (1967), 242–245.Google Scholar
3. Giffen, C. H., Desuspendability of free involutions on Brieskorn spheres, Bull. Amer. Math. Soc. 75 (1969), 426–429.Google Scholar
4. Hirzebruch, F. and Mayer, K. H., 0﹛n)-Mannigfaltigkeiten exotische Sphàren una Singularitdten, Lecture Notes in Mathematics, No. 57 (Springer-Verlag, Berlin, 1968).Google Scholar
5. Kervaire, M. A. and Milnor, J. W., Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537.Google Scholar
6. Lopez de Medrano, S., Some results on involutions of homotopy spheres, Proceedings of the Conference on Transformation Groups, pp. 167–174 (Springer-Verlag, Berlin, 1968).Google Scholar
7. Orlik, P., Seminar notes on simply connected surgery (Institute for Advanced Study, Princeton, N.J., 1968).Google Scholar
8. Orlik, P. and Rourke, C. P., Free involutions on homotopy spheres (in preparation).Google Scholar
9. Wall, C. T. C., Freepiecewise linear involutions on spheres, Bull. Amer. Math. Soc. 74 (1968), 554–558. 1Q# Surgery of compact manifolds, preprint.Google Scholar