An ensemble of cosmological models based on generalized BFtheory is constructed where the rôle of vacuum (zero-level) coupling constants is played by topologically invariant rational intersection forms (cosmological-constant matrices) of 4-dimensional plumbed Vcobordisms which are interpreted as Euclidean spacetime regions. For these regions describing topology changes, the rational and integer intersection matrices are calculated. A relation is found between the hierarchy of certain elements of these matrices and the hierarchy of coupling constants of the universal (low-energy) interactions.

We consider an Abelian BF-model in the frame of ten-dimensional Kaluza–Klein approach on the space T2×X×M, where X belongs to the class of four-dimension decorated plumbed cobordisms (dp-cobordisms) and M is an An-1-singularity resolution manifold homeomorphic to a compactified ALE space. These four-dimensional manifolds with boundaries possess nontrivial cohomology properties that lead to a specific generalization of the Dirac quantization conditions and enables us to express classical partition functions in terms of 4-form fluxes through the direct product of nontrivial 2-cycles associated with the manifolds X and M. The intersection matrices of these manifolds play the role of coupling constants for the fluxes. We build several examples of dp-cobordisms containing in their intersection matrices the hierarchy of dimensionless low-energy coupling constants of interactions which are available in the real universe. We also consider the phenomenon of "running coupling constants," in particular the cosmological constant evolution induced by the topology changes of internal space X.

We consider the block matrices and 3-dimensional graph manifolds associated with a special type of tree graphs. We demonstrate that the linking matrices of these graph manifolds coincide with the reduced matrices obtained from the Laplacian block matrices by means of Gauss partial diagonalization procedure described explicitly by W. Neumann. The linking matrix is an important topological invariant of a graph manifold which is possible to interpret as a matrix of coupling constants of gauge interaction in Kaluza-Klein approach, where 3-dimensional graph manifold plays the role of internal space in topological 7-dimensional BF theory. The Gauss-Neumann method gives us a simple algorithm to calculate the linking matrices of graph manifolds and thus the coupling constants matrices.

The hierarchy and fine tuning of the gauge coupling constants are described on the base of topological invariants (Chern classes interpreted as filling factors) characterizing a collection of fractional topological fluids emerging from three dimensional graph manifolds, which play the role of internal spaces in the Kaluza-Klein approach to the topological BF theory. The hierarchy of BF gauge coupling constants is simulated by diagonal elements and eigenvalues of rational linking matrices of tree graph manifolds pasted together from Brieskorn (Seifert fibered) homology spheres. Specific examples of graph manifolds are presented which contain in their linking matrices the hierarchy of coupling constants distinctive for the dimensionless coupling constants in our Universe. The fine tuning effect is simulated owing to the special numerical properties of diagonal elements of the linking matrices. We pay a particular attention to fine tuning problem for the cosmological constant.

A model of topological field theory is presented in which the vacuum coupling constants are topological invariants of the four-dimensional spacetime. Thus the coupling constants are theoretically computable, and they indicate the topological structure of our universe.We construct an Abelian BF -type model in analogy with the ordinary fourdimensional topological field theory 1 and with the low-energy effective U (1) r -theory of Seiberg-Witten (SW) 2 , beginning with a U (1) r -bundle E over a four-dimensional topological space X with a non-empty boundary ∂X, E being a direct sum of linear bundles L 1 ⊕ · · · ⊕ L r . Let us define locally connection 1-forms A a (a = 1, . . . , r) on E with values in the algebra L of the group U (1), and 2-forms B a with values in the dual algebra. Due to these analogies it is natural to write the action aswhere F a = dA a ; Λ ab and Θ ab are non-degenerate symmetric matrices called those of the coupling constants and theta angles matrices, respectively. Our action admits symmetry under dual conjugation similar to the electro-magnetic one (EM duality) of the SW theory,These duality transformations carry S into its equivalent, S D , and they involve the strong-weak coupling duality. 2 We call Λ ab and Λ ab strong and weak coupling constants matrices (in this sense), respectively.Then we generalize Dirac's quantization conditions: the flux through non-trivial 2-cycles Σ I must be ΣI F a = 2πm a I , m a I ∈ Z. Then from dynamical equations of our BF system dB a = 0, F a = Λ ab B b together with the gauge symmetries it follows that the moduli space of this BF system is H 2 (X, Z) ⊕ H 2 (X, ∂X, Z) meaning that [ 1 2π F a ] ∈ H 2 (X, Z) and [ 1 2π B a ] ∈ H 2 (X, ∂X, Z). Thus there are no local degrees of freedom and like in the case of the low energy effective SW action, our model describes the moduli space of vacua. The spacetime topology is non-trivial since we model the spacetime by the graph manifold. 3 Each tree graph corresponds to a unique four-dimensional space X with a boundary containing lens spaces and Z-homology spheres. The latter ones are results of splicing of Seifert fibred homology (Sfh) spheres. The most important construction element is Sfh-sphere Σ(a) ≡ Σ(a 1 , a 2 , a 3 ) having three special orbits and being a three-dimensional manifold which is an intersection of the Brieskorn surface z 1 a1 + z 2 a2 + z 3 a3 = 0 (z i ∈ C i ) and a sphere S 5 . Here a 1 , a 2 and a 3 are mutually prime integers (Seifert invariants). To the end of constructing our cosmological model we need a specific family of Sfh-spheres to which we give the following definition: 3 We take a succession of Sfh-spheres calling it the primary one: {Σ(q 2n−1 , p 2n , p 2n+1 )|n = 0, . . . , 4}. Here p i is the i-th prime number in the natural series and q i = p 1 · · · p i . Then we define the "derivative" of a Sfh-sphere Σ(a)