Classifies smooth closed oriented manifolds homotopy equivalent to three families of sphere products up to almost diffeomorphism by realizing the image of the normal-invariant map with explicit constructions from bundles and plumbings.
Finite group actions on Kervaire manifolds
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abstract
The (4k+2)-dimensional Kervaire manifold is a closed, piecewise linear (PL) manifold with Kervaire invariant 1 and the same homology as the product of two (2k+1)-dimensional spheres. We show that a finite group of odd order acts freely on a Kervaire manifold if and only if it acts freely on the corresponding product of spheres. If the Kervaire manifold M is smoothable, then each smooth structure on M admits a free smooth involution. If k + 1 is not a 2-power, then the Kervaire manifold in dimension 4k+2 does not admit any free TOP involutions. Free "exotic" (PL) involutions are constructed on the Kervaire manifolds of dimensions 30, 62, and 126. Each smooth structure on the 30-dimensional Kervaire manifold admits a free Z/2 x Z/2 action.
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Smooth manifolds homotopy equivalent to products of spheres
Classifies smooth closed oriented manifolds homotopy equivalent to three families of sphere products up to almost diffeomorphism by realizing the image of the normal-invariant map with explicit constructions from bundles and plumbings.