  We revisit an argument due to Lesch (Topology 32 (1993), no. 3, 611-623) for
proving the cobordism invariance of the index of Dirac operators on
even-dimensional closed manifolds and combine this with recent work by the
author (New York J. Math. 28 (2022), 705-772) to show vanishing results for the
spectral flow for families of selfadjoint Fredholm realizations of elliptic
operators in case the family is induced on the boundary by an elliptic operator
on a compact space. This work is motivated by studying the behavior of the
index of realizations of elliptic operators under cobordisms of statified
manifolds.
