2014/11/26 by Zadeh, Mostafa E., Moghadasi, Reza
#53C99 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1411.7259
It is well known that the cohomology groups of a closed manifold M can be reconstructed using the gradient dynamical of a Morse-Smale function f\colon M→ \R. A direct result of this construction are Morse inequalities that provide lower bounds for the number of critical points of f in term of Betti numbers of M. These inequalities can be deduced through a purely analytic method by studying the asymptotic behaviour of the deformed Laplacian operator. This method was introduced by E. Witten and has inspired a numbers of great achievements in Geometry and Topology in few past decades. In this paper, adopting the Witten approach, we provide an analytic proof for; the so called; equivariant Morse inequalities when the underlying manifold is acted on by the Lie group G=S1 and the Morse function f is invariant with respect to this action.