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Description

4-dimensional surgery is a fundamental technique underlying geometric classification results for topological 4 manifolds. It is known to work in the topological category for a class of "good" fundamental groups. This result was originally established in the simply-connected case by Freedman in 1981, and it is currently known to hold for groups of sub exponential growth and a somewhat larger class generated by these. The A-B slice problem is a reformulation of the surgery conjecture for free groups, which is the most difficult case. In this talk I will show that the A-B slice problem admits a link-homotopy+ solution. The proof relies on geometric applications of the group-theoretic 2-Engel relation. I will also discuss implications for the surgery conjecture. (Joint work with Mike Freedman)