Local and global topological complexity measures OF ReLU neural network functions

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.
Comment: 40 pages, 8 figures; Sections 5 and 6 from v1 removed from v2. We plan to use constructions in those sections for a follow-up paper with a more computational focus