Topological 4manifolds with 4dimensional fundamental group
Abstract
Let $\pi$ be a group satisfying the FarrellJones conjecture and assume that $B\pi$ is a 4dimensional Poincaré duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$ has degree 1 and show that two such manifolds are scobordant if and only if their equivariant intersection forms are isometric and they have the same KirbySiebenmann invariant. If $\pi$ is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same KirbySiebenmann invariant. This shows rigidity in many cases that lie between aspherical 4manifolds, where rigidity is expected by Borel's conjecture, and simply connected manifolds where rigidity is a consequence of Freedman's classification results.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 arXiv:
 arXiv:2007.03399
 Bibcode:
 2020arXiv200703399K
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Algebraic Topology;
 57K40;
 57N65
 EPrint:
 Results are generalized to the nonorientable case, to appear in Glasgow Math. J., 8 pages