Asian Journal of Mathematics

Volume 27 (2023)

Number 1

Simply and tangentially homotopy equivalent but non-homeomorphic homogeneous manifolds

Pages: 57 – 76

DOI: https://dx.doi.org/10.4310/AJM.2023.v27.n1.a2

Author

Sadeeb Simon Ottenburger (Institute for Thermal Energy Technology and Safety (ITES), Karlsruhe Institute of Technology (KIT), Eggenstein-Leopoldshafen, Germany)

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 nonhomeomorphic $5$-dimensional closed homogeneous spaces with fundamental group isomorphic to $\mathbb{Z}/r$. 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. As a by-product, if $L$ is a smooth and closed manifold homotopy equivalent to the standard $3$-dimensional lens space then the moduli space of complete smooth metrics of nonnegative sectional curvature on $L \times S^2 \times \mathbb{R}$ has infinitely many components. 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.

Keywords

homogeneous manifolds, homotopy classification, surgery, non-negative curvature, pinching, souls

2010 Mathematics Subject Classification

53C21, 53C30, 57R20, 57R22, 57R55

Received 24 April 2021

Accepted 7 December 2022

Published 16 June 2023