The Morse inequalities for line bundles
Samir A. Khabbaz · Journal of the Australian Mathematical Society · 1970
In place of a real valued differentiable (C2) function on a closedn-dimensional differentiable manifoldM, we may more generally consider a differentiable section s in any line bundleLonM, assumed to have structural groupZ2, the group of integers modulo two. Since the usual definitions of a critical point and of a non-degenerate critical point are local in nature, and since composing a differentiable real valued function with the functiont→—t does not change its set of critical points or its set of non-degenerate critical point, it is clear that we may speak of critical points and nondegenerate critical points of the section s. Unless the bundle has a fixed trivialization however, the index of a non-degenerate critical point must be thought of as a set of two numbers {k, n—k), corresponding to the two indices arising from the two trivializations possible forLrestricted to a small enough neighborhood of the point, i.e. corresponding to the two possible ways of reading the index. With this understanding we extend the usual definitions, and call a differentiable (C2) section s of L a Morse section if each of its critical points is non-degenerate.