Geodesic
On the ARF Invariant of an Involution
Canadian journal of mathematics, Tome 22 (1970) no. 3, pp. 519-524

Voir la notice de l'article provenant de la source Cambridge University Press

Let Σ 4k+1 denote a smooth manifold homeomorphic to the (4k + 1)-sphere, S 4k+1k ≧ 1, and T: Σ4k+1 → Σ 4k+1 a differentiate free involution. Our aim in this note is to derive a connection between the differentiate structure on Σ 4k+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 S 4k+1, k ≧ 1, form a finite abelian group θ 4k+1 with group operation connected sum. The elements are called homotopy spheres. Those homotopy spheres that bound parallelizable manifolds form a subg roup bP 4k+2 ⊂ θ4k+1. It is proved in [5, Theorem 8.5] that bP 4k+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. 
Orlik, Peter. On the ARF Invariant of an Involution. Canadian journal of mathematics, Tome 22 (1970) no. 3, pp. 519-524. doi: 10.4153/CJM-1970-060-8
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