  Topological insulators are a novel state of matter that share a common
feature: their spectral bands are associated with a nonlocal integer-valued
index, commonly manifesting through quantized bulk phenomena and robust
boundary effects. In this work, we demonstrate using dimensional reduction that
high-order topological insulators are descendants from a chiral semimetal in
higher dimensions. Specifically, we analyze the descendants of an ancestor
four-dimensional Chern insulator in the limit where it becomes chiral and show
their relation to two-dimensional second-order topological insulators.
Correspondingly, the quantization of the charge accumulation at the corners of
the 2D descendants is obtained and related to the topological indices -- the
1st and 2nd Chern numbers -- of the ancestor model. Our approach provides a
connection between the boundary states of high-order topological insulators and
topological pumps -- the latter being dynamical realizations of
high-dimensional Chern insulators.
