4 On the Metric Independent Exotic Homology

Different types of nonstandard homology groups based on the various subcomplexes of differential forms are considered as a continuation of the recent authors works. Some of them reflect interesting properties of dynamical systems on the compact manifolds. In order to study them a Special Perturbation Theory in the form of Spectral Sequences is developed. In some cases a convenient fermionic formalism of dealing with differential forms is used originated from the work of Witten in the Morse Theory (1982)and the authors work where some nonstandard analog of Morse Inequalities for vector fields was found (1986). In the work [10] we invented some sort of exotic homology of the first and second kind. Let us remind here that the second kind exotic homology defined in this work are constructed in the following natural way: for every linear space L with operator d ′ : L → L we can define homology on any d-invariant subspace T ⊂ L such that (d ′) 2 : T → 0. So we have by definition H T = Ker(d ′)/Im(d ′). Our main example was based on the De-Rham complex Λ * (M) = n i=0 Λ i of real-valued differential forms for any C ∞-manifold M. We considered a family of operators d ′ = d+λω * such that d ′ (a) = da+λω∧a. Here ω is an one-form (maybe non closed), and a is any C ∞-form. A lot of work was done since 1986 (see, for example, [1, 2]) for the case of the closed one-form ω where (d ′) 2 = 0. We do not discuss this case here. I.Perturbation of d by the nonclosed odd form.