Journal of the
Korean Mathematical Society JKMS

The classical Morse inequality gives a relation between the number of critical points of a Morse function on a compact manifold $M$ and the Betti numbers of $M$. Betti numbers are the dimensions of the cohomology spaces of the trivial line bundle on $M$. We consider an arbitrary flat vector bundle $E$ over $M$, which gives cohomology spaces $H^k(M,E)$, and show that a similar Morse inequality holds with Betti numbers of $M$ replaced by the dimensions of the cohomology spaces $H^k(M,E)$.

Keywords: Morse inequality, flat bundles, elliptic operators, connections

MSC numbers: 53C05, 58E05