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The Morse-Bott inequalities relate the topology of a closed manifold to thetopology of the critical point set of a Morse-Bott function defined on it. TheMorse-Bott inequalities are sometimes stated under incorrect orientationassumptions. We show that these assumptions are insufficient with an explicitcounterexample and clarify the origin of the mistake.

The Morse-Bott inequalities, orientations, and the Thom isomorphism in Morse homology