Thom-Milnor bounds for smooth manifolds

We prove a smooth analogue of the classical Thom-Milnor bound, showing that the Betti numbers of the zero set of a smooth map on a compact Riemannian manifold can be controlled by a condition number computed from its first jet. This extends previous results in the Euclidean setting by Lerario and Stecconi [J. Singul., 2021]. As a key step, we generalize the Thom-Milnor bound to polynomial maps on a nonsingular real algebraic variety, improving the dependence on the degree. Finally, inspired by the work of B\"{u}rgisser, Cucker and Tonelli-Cueto [Found. Comput. Math., 2020], we introduce a condition number for families of functions. Using this we extend existing bounds due to Basu, Pollack and Roy [Proc. Amer. Math. Soc., 2004], for the Betti numbers of semialgebraic sets described by closed conditions to what we call closed semialgebraic type sets, namely sets defined by closed inequalities involving smooth functions.