Suppose given a regular covering space over a closed manifold, with discrete group Γ: (1) pi: M ̂ →Mn, γ: M ̂ → M̂, γ ∈ Γ. The Generalized Morse Problem (Problem 1). Find effective lower bounds

on the number of critical points of a smooth function f on Mn, taking account of the group Γ (see [1], where the question of existence of such bounds is examined, but not effectively). Problem 2 (the author [2, 3]). Find lower bounds on the number of critical points of a closed 1-form ω on Mn. Of particular interest in Problem 2 is the case Γ = Zk, where k is the number of integrals of ω, over 1-cycles in Mn, that are linearly independent over Z. The problem was solved by the author in [2] and [3] for the case k = 1, and it was shown by Farber in [4] that the bounds in [2] are sharp. Here we shall examine only those bounds that can theoretically be obtained from the properties of elliptic operators on the manifold, by analogy with [5]. Every representation (2) ρ: Γ → GL(N,C) determines a complex of forms Λ∗ρ, where (3) Ω ∈ Λ∗ρ ⊂ Λ∗(M̂,CN), γ∗Ω = ρ(γ)Ω, γ ∈ Γ. The cohomology of the complex (Λ∗ρ, d) coincides with the cohomologyH ρ (M n,CN) with local coefficients (see [6], p. 134). We have the obvious Proposition 1. For any Morse function f on Mn, (4) mj(f)> ma