Smooth Structures on Eschenburg Spaces: Numerical Computations

Abstract

This paper numerically computes the topological and smooth invariants of Eschenburg–Kruggel spaces with small fourth cohomology group, following Kruggel’s determination of the Kreck–Stolz invariants of Eschenburg spaces satisfying condition C. It is shown that each topological Eschenburg–Kruggel space with small fourth cohomology group has each of its 28 oriented smooth structures represented by an Eschenburg–Kruggel space. Our investigations also suggest that there is an action of $\mathbb{Z}_{12}$ on the set of homotopy classes of Eschenburg–Kruggel spaces, the nature of which remains to be understood.

The calculations are done in C++ with the GNU GMP arbitrary precision library and Jon Wilkening’s C++ wrapper.