Title:  L^2 methods, knot concordance, localization and amenable groups.


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Abstract:  L^2 signatures play a central role in the study of knot concordance, a classical relation on knots closely allied with deep considerations in singularity theory and the classification of 4-manifolds.  Using a new approach which subsumes past results, we extend the above techniques to the related problem of classifying manifolds up to homology cobordism, and significantly extend key results concerning invariance of L^2 signatures and betti numbers.  We exhibit new examples of homology equivalent manifolds in low and high dimensions which are not homology cobordant.  Many of these results involve groups with torsion, unassailable via prior tools.