  We apply a generalized piecewise-linear (PL) version of Morse theory due to
Grunert-Kuhnel-Rote to define and study new local and global notions of
topological complexity for fully-connected feedforward ReLU neural network
functions, F: R^n -> R. Along the way, we show how to construct, for each such
F, a canonical polytopal complex K(F) and a deformation retract of the domain
onto K(F), yielding a convenient compact model for performing calculations. We
also give a construction showing that local complexity can be arbitrarily high.
