Abstract
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.
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