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Geometry and Topology of Some Fake Projective Spaces

  • Author(s): Rajan, Priyanka;
  • Advisor(s): Wilhelm, Fred;
  • et al.
Abstract

A fake real projective space is a manifold homotopy equivalent to real projective space, but not diffeomorphic to it. Equivalently, it is the orbit space of a free involu- tion on a (homotopy) sphere.

In this thesis, we show that some of the fake RP6s , constructed by Hirsch and Milnor in 1963, and the analogous fake RP14s admit metrics that simutaneously have almost nonnegative sectional curvature and positive Ricci curvature. These spaces are obtained by taking the Z2 quotients of the embedded images of the standard spheres of codimension one in some of Milnor’ s exotic 7−spheres and the analogous Shimada’s exotic 15−spheres. This part of my thesis is joint work with F. Wilhelm.

Hirsch and Milnor also constructed fake RP5s using invariant subspheres of codi- mension two. Octonionically, this construction yields closed 13−manifolds, that are homotopy equivalent to RP13s. The analog to their proof that fake RP5s are not diffeomorphic to standard RP5 breaks down; since in contrast to dimension 6, there is an exotic 14−sphere. We show that some of the Hirsch-Milnor RP13s are not diffeomorphic to standard RP13s. Here we obtain a complete diffeomorphism clas- sification of the Hirsch-Milnor RP13s. This part of my thesis is joint work with C. He.

Using techniques of group diagrams we provide an elementary proof that certain 3-dimensional Brieskorn varieties are SO(2) × SO(2)−equivariantly diffeomorphic to certain Lens Spaces.

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