The wikipedia page on 'Algebraic Topology' contains the following sentence:
One can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf cohomology to investigate the solvability of differential equations defined on the manifold in question.
Unfortunately, no concrete references are added.
Are there good textbooks or other sources on applications of algebraic topology to solvability (and perhaps other qualitative aspects) of differential equations?
Let me suggest a slim book by Charles Conley entitled "Isolated invariant sets and the Morse index", published in 1978. The Conley index was born in this book and has been an important method in the application of algebraic topology to dynamical systems every since.