  The main goal of this paper is to give a unified treatment to many known
cuplength estimates. As the base case, we prove that for $C^0$-perturbations of
a function which is Morse-Bott along a closed submanifold, the number of
critical points is bounded below in terms of the cuplength of that critical
submanifold. As we work with rather general assumptions the proof also applies
in a variety of Floer settings. For example, this proves lower bounds for the
number of fixed points of Hamiltonian diffeomorphisms, Hamiltonian chords for
Lagrangian submanifolds, translated points of contactomorphisms, and solutions
to a Dirac-type equation.
