# Topological rigidity for non-aspherical manifolds

@article{Kreck2005TopologicalRF,
title={Topological rigidity for non-aspherical manifolds},
author={M. Kreck and W. Lueck},
journal={Pure and Applied Mathematics Quarterly},
year={2005},
volume={5},
pages={873-914}
}
• Published 2005
• Mathematics
• Pure and Applied Mathematics Quarterly
The Borel Conjecture predicts that closed aspherical manifolds are topological rigid. We want to investigate when a non-aspherical oriented connected closed manifold M is topological rigid in the following sense. If f : N ! M is an orientation preserving homotopy equivalence with a closed oriented manifold as target, then there is an orientation preserving homeomorphism h: N ! M such that h and f induce up to conjugation the same maps on the fundamental groups. We call such manifolds Borel… Expand
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#### References

SHOWING 1-10 OF 53 REFERENCES
Rigidity and other topological aspects of compact nonpositively curved manifolds
• Mathematics
• 1990
Let M be a compact connected Riemannian manifold whose sectional curvature values are all nonpositive. Let T denote the fundamental group of M. We prove that any homotopy equivalence ƒ : N —> M fromExpand
A topological analogue of Mostow’s rigidity theorem
• Mathematics
• 1989
Three types of manifolds, spherical, fiat, and hyperbolic, are paradigms of geometric behavior. These are the Riemannian manifolds of constant positive, zero, and negative sectional curvatures,Expand
On Simply-Connected 4-Manifolds
This paper concerns (but does not succeed in performing) the diffeomorphism classification of closed, oriented, differential, simply-connected 4-manifolds. It arises out of the observation (due toExpand
On Invariants of Hirzebruch and Cheeger{Gromov
• Mathematics
• 2003
We prove that, if M is a compact oriented manifold of dimension 4k +3 , where k> 0, such that 1(M) is not torsion-free, then there are innitely many manifolds that are homotopic equivalent to M butExpand
Surgery and duality
Surgery, as developed by Browder, Kervaire, Milnor, Novikov, Sullivan, Wall and others is a method for comparing homotopy types of topological spaces with difieomorphism or homeomorphism types ofExpand
On the cut-and-paste property of higher signatures of a closed oriented manifold
• Mathematics
• 2000
Abstract We extend the notion of the symmetric signature σ( M ,r)∈L n (R) for a compact n-dimensional manifold M without boundary, a reference map r : M→BG and a homomorphism of rings withExpand
On 3-manifolds
It is well known that a three dimensional (closed, connected and compact) manifold is obtained by identifying boundary faces from a polyhedron P. The study of (\partial P)/~, the boundary \partial PExpand
Algebraic L-theory and Topological Manifolds
Introduction Summary Part I. Algebra: 1. Algebraic Poincare complexes 2. Algebraic normal complexes 3. Algebraic bordism categories 4. Categories over complexes 5. Duality 6. Simply connectedExpand
Surgery on compact manifolds
Preliminaries: Note on conventions Basic homotopy notions Surgery below the middle dimension Appendix: Applications Simple Poincare complexes The main theorem: Statement of results An importantExpand
The Kervaire invariant of framed manifolds and its generalization
In 1960, Kervaire [11] introduced an invariant for almost framed (4k + 2)manifolds, (k # 0, 1, 3), and proved that it was zero for framed 10-manifolds, which was a key step in his construction of aExpand