4-dimensional $L^2$-acyclic bordism and Whitney towers

Recently Sylvain Cappell, Jim Davis and Shmuel Weinberger studied $L^2$-acyclic bordism groups in high dimensions.  In this talk we address 4-dimensional $L^2$-acyclic bordism between 3-manifolds.  We introduce a Whitney tower approach to study the structure in this dimension, and show that Cheeger-Gromov invariants over amenable groups give obstructions.  Also we answer some questions of Cappell, Davis and Weinberger which concern the relationship of $L^2$-acyclic bordism and knot concordance.