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 [Eschenburg 82, Kruggel 05, Kreck and Stolz 91]. 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 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.

2000 AMS Subject Classification:

• Primary 57R55
• Secondary 53C25

Keywords:

• Smooth structures
• Eschenburg spaces
• Gnu Gmp

ACKNOWLEDGMENTS

The author thanks T. Chinburg, C. Escher, and W. Ziller for helpful comments and providing their Maple code; J. Wilkening for releasing his Gmpfrxx code; M. Kreck for discussions of [Kruggel 05]; and C. Peterson.

Notes

¹In [Butler 09, Tables 5–Tables 7], one finds that the sums reported lie in [−2, 2]. Those spaces with sum reported in [−2, −1] are obtained by reversing the orientation of a space whose sum lies in [1, 2].

²Available at http://gmplib.org/ .

³At http://math.berkeley.edu/wilken/code/gmpfrxx/ .

⁴In replicating these results, differing conventions for the projection map became apparent. The Chinburg–Escher–Ziller code uses the convention that x is reduced modulo 1, and then is mapped to itself by the identity and is mapped to by a constant shift. In our C++ code, x is reduced modulo 1, then shifted by .