Simply and tangentially homotopy equivalent but non-homeomorphic homogeneous manifolds

Abstract

For each odd integer r greater than one and not divisible by three we give explicit examples of infinite families of simply and tangentially homotopy equivalent but pairwise non-homeomorphic closed homogeneous spaces with fundamental group isomorphic to Z/r. As an application we construct the first examples of manifolds which possess infinitely many metrics of nonnegative sectional curvature with pairwise non-homeomorphic homogeneous souls of codimension three with trivial normal bundle, such that their curvatures and the diameters of the souls are uniformly bounded. These manifolds are the first examples of manifolds fulfilling such geometric conditions and they serve as solutions to a problem posed by I. Belegradek, S. Kwasik and R. Schultz.

On the fibre structure of compact homogeneous spaces

“…However, it turns out that there are compact homogeneous spaces with a finite fundamental group that are not homeomorphic, although they are simply (we are talking about the concept of simple homotopy equivalence) and tangentially homotopy equivalent. Manifolds of dimension 5 of this kind are given in [24]. They are spaces of bundles over a two-dimensional sphere in which the fibre is a lens space.…”

“…In fact, the semisimple Lie group SU(2) × SU( 2) is also transitive on these manifolds. The point is that the varieties under consideration are compact, have a finite fundamental group, and therefore, by virtue of the very general Montgomery theorem (see, for example, [1]) the maximal semisimple subgroup of a transitive Lie group is also transitive on this variety (in the examples from [24] this will be the group SU(2) × SU(2) -it is easy to find out even by a detailed study of the transitive action described there).…”

“…We use this fact to prove the possibility of non-uniqueness of the fibre of the natural bundle for compact homogeneous spaces. To do this, take some two (denoted N 5 1 , N 5 2 ) from five-dimensional manifolds constructed in [24] which are not homeomorphic, but are tangentially homotopy equivalent. Consider the direct products of these two manifolds by the torus T (although here they are usually multiplied by an open disc, but in fact only the parallelizability of this factor is important) of a sufficiently large dimension (it is enough to take this dimension by 2 greater than the dimension of the multiplied manifolds).…”