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.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/158