  We apply the topological theory of symmetry indicators to interaction-induced
exciton band structures in centrosymmetric semiconductors. Crucially, we
distinguish between the topological invariants inherited from the underlying
electron and hole bands, and those that are intrinsic to the exciton
wavefunction itself. Focusing on the latter, we show that there exists a class
of exciton bands for which the maximally-localised exciton Wannier states are
shifted with respect to the electronic Wannier states by a quantised amount; we
call these excitons shift excitons. Our analysis explains how the exciton
spectrum can be topologically nontrivial and sustain exciton edge states in
open boundary conditions even when the underlying noninteracting bands have a
trivial atomic limit. We demonstrate the presence of shift excitons as the
lowest energy neutral excitations of the Su-Schrieffer-Heeger model in its
trivial phase when supplemented by local two-body interactions, and show that
they can be accessed experimentally in local optical conductivity measurements.
