  We analyze the topology and dispersion of bulk Wannier excitons in nanosheets
of topological insulators in the family of bismuth chalcogenides. Our main
finding is that excitons also inherit the topology of the electronic bands,
quantified by the skyrmion winding numbers of the constituent electron and hole
pseudospins as a function of the total exciton momentum. We furthermore show
that every $s$-wave exciton state consists of a quartet with a degenerate and
quadratically dispersing nonchiral doublet, and a chiral doublet with one
linearly dispersing mode as in transition metal dichalcogenides. We also
demonstrate the existence of topological edge states of chiral excitons arising
from the bulk-boundary correspondence.
