Hirzebruch-type invariants of Cheeger-Gromov and low dimensional topology

A signature defect invariant of Hirzebruch over the group of order two was generalized to more general cases by Atiyah-Patodi-Singer and Cheeger-Gromov.  I will discuss some recent uses of invariants of this type in low dimensional topology, including smooth/topological disk embedding in dimension 4, and quantitative understanding of 3-manifolds.