Celebratio Mathematica — Freedman

The A-B slice problem

Works connected to Peter Teichner

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article M. H. Freedman and P. Teichner: “4-manifold topology I: Subexponential groups,” Invent. Math. 122 : 3 (1995), pp. 509–529. MR 1359602 Zbl 0857.57017

The technical lemma underlying the 5-dimensional topological \( s \)-cobordism conjecture and the 4-dimensional topological surgery conjecture is a purely smooth category statement about locating \( \pi_1 \)-null immersions of disks. These conjectures are theorems precisely for those fundamental groups (“good groups”) where the \( \pi_1 \)-null disk lemma (NDL) holds. We expand the class of known good groups to all groups of subexponential growth and those that can be formed from these by a finite number of application of two operations: (1) extension and (2) direct limit. The finitely generated groups in thisclass are amenable and no amenable group is known to lie outside this class.