I'm currently learning some Morse theory, and I've just learned about the Morse inequalities. They appear to be very powerful - being able to deduce so much topological information from a single real valued function on the manifold is pretty mindblowing.

That being said, I haven't managed to find any interesting corollaries or applications of the Morse inequalities (beyond obvious calculations like, any Morse function on a torus must have at least 4 critical points etc). I realise this question is very open, but are people familiar with any results that either crucially use the Morse inequalities, or become much easier to prove using the Morse inequalities?

  • $\begingroup$ maybe they arent useful $\endgroup$ – Sandeep Silwal Mar 28 at 23:12

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