Morse inequalities and von Neumann II1factors

In this paper it is shown that in the Morse inequalities on a multiply connected manifold, instead of the classical Betti numbers, one can consider real Betti numbers equal to the von Neumann relative dimensions of the homology groups or cohomol-ogy groups of the local system obtained from the representation of the fundamental group in a II1-factor, according to the conjecture of [1]. The reason for this consists in the following. Even in finite-dimensional representations χ: pi1(X) → GL(N,C), the analogues of the Morse inequalities for a function f on a smooth n-dimensional closed manifold X have the form (1) Nmp(f)> bχp, where mp(f) is the number of critical points of the index p of f and bχρ = b χ p (X) is the dimension of the space of p-dimensional homology of X with values in CN as in a pi1-module with representation χ. Inequality (1) follows from the fact that its left-hand side is in fact the dimension of the group of chains in the comple