Manifolds Homotopy Equivalent to Certain Torus Bundles over Lens Spaces

Abstract

We compute the topological simple structure set of closed manifolds that occur as total spaces of flat bundles over lens spaces S l =.Z=p/ with fiber T n for an odd prime p and l 3 provided that the induced Z=p-action on 1 .T n / D Z n is free outside the origin. To the best of our knowledge this is the first computation of the structure set of a topological manifold whose fundamental group is not obtained from torsionfree and finite groups using amalgamated and HNNextensions. We give a collection of classical surgery invariants such as splitting obstructions and -invariants that decide whether a simple homotopy equivalence from a closed topological manifold to M is homotopic to a homeomorphism. © 2020 Wiley Periodicals LLC Contents 0. Introduction 1. Preliminaries about the Group Z n Ì Z=p 2. Preliminaries about the Farrell-Jones Conjecture 3. Algebraic K-theory of the Group Ring 4. Algebraic L-theory of the Group Ring 5. Structure Sets 6. Periodic Simple Structure Set of BP for a Finite p-group 7. Periodic Simple Structure Set of B 8. Periodic Simple Structure Set of M 9. Geometric Simple Structure Set of M 10. Invariants for Detecting the Structure Set of M 11. Appendix: Open Questions Bibliography

Torus bundles over lens spaces

“…Proof. The proof is similar to the proof of [DL,Lemma 8.3] so we give an outline. We first show that the differentials in the first spectral sequence vanish.…”