For questions about Morse theory, which is a branch of differential topology to analyze topology of manifolds by studying differentiable functions on them.
In differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold.
According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology.
Together with differential-topology if necessary.