Čech-de Rham Cohomology of a Refinement of a Principal Bundle

Gheorghe Ivan

Abstract

In this paper we shall study the cohomology of the various spaces appearing in the refinement of a differentiable principal bundle defined by a closed subgroup.

Mathematica Subject Classification: 55R45 Key words: tissue associated to a fibre bundle, refinement of a principal bundle, Čech-de Rham complex, Čech-de Rham cohomology

Introduction

Let p : E ⟶ B be a differentiable principal bundle and let N_{q} = (G = F_{0} ⊃ F_{1} ⊃ ... ⊃ F_{q} = {e}) (q is an integer ≥ 2) be a sequence of closed subgroups of G.Let E_{i} = E/F_{i}, i=0,1,...,q; F_{k}^{j} = F_{j}/F_{k}, 0 ≤ j < k ≤ q; G_{k}^{j} = F_{j}/N_{jk} (here N_{jk} is the normal closure of F_{j} in F_{k}).Finally, let p_{jk} : E_{k} ⟶ E_{j} be the canonical map.

D.I. PAPUC ([5]) proved that p_{jk} : E_{k} ⟶ E_{j} is a differentiable fibre bundle with fibre F_{k}^{j} and structure group G_{k}^{j}.

A refinement of a principal bundle ξ = (E, p, B, G) is the well-known structure determined by a closed subgroup F_{j} of G constitued by three bundles (ξ; ξ_{oj}, ξ_{jq}).

The paper consists of three sections. The first section contains some preliminaries about the tissues associated to a principal bundle. Also, some examples of tissues and refinements are given.

The second section one contains the construction of the Čech-de Rham complex of an open cover of a manifold (see, [3]).

In the third section we shall study the Čech-de Rham cohomology of a refinement of a principal fibre bundle, whenever the base space of tissue has a finite good cover. Some main results concerning the cohomology of the spaces appearing in this structure are established.

Throughout in this note all spaces are finite-dimensional real differentiable manifolds, without boundary of C^{∞} classes and all maps are C^{∞}.

Balkan Journal of Geometry and Its Applications, Vol.2, No.2, 1997, pp. 69-78 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

1 Refinements of a differentiable principal bundle

Let (ξ, N_{⨿}) be a pair consisting of a differentiable principal Steenrod bundle ξ = (E, p.B, G; A) and N_{⨿} = (G = F_{′} ⊃ F_{∞} ⊃ ... ⊃ F_{⨿−∞} ⊃ F_{⨿} = {⌉}) a sequence of closed subgroups of the structure group G.

We consider the Steenrod bundles ξ_{jk} = (E_{k}, p_{jk}, E_{j}, F_{k}^{j}, G_{k}^{j}; A_{|∥}) for 0 ≤ j < k ≤ q determined by ξ and N_{⨿}, where E_{j} = E/F_{j}, F_{k}^{j} = F_{j}/F_{k}, G_{k}^{j} = F_{j}/N_{jk}.N_{jk} being the largest normal subgroup of F_{j} included in F_{k} and p_{jk} is the canonical map, (see [5],p.372).

The Steenrod tissue [ξ, N_{⨿}] associated to the pair (ξ, N_{q}) is the set of all fibre bundles ξ_{jk} for 0 ≤ j < k ≤ q. We have that ξ_{oq} = ξ and moreover every fibre bundle ξ_{jq},0 < j < q is a principal one.

The triple (ξ = ξ_{oq}; ξ_{0j}, ξ_{jq}), for 0 < j < q is called, via [5], the refinement of ξ defined by F_{j}. Example 1. a) The tissue associated to bundle of tangent linear frames. Let M be a manifold of dimension n. A k-tangent linear frame u_{k} at a point x ∈ M, where 1 ≤ k ≤ n is a linear independent system u_{k} = (X_{1}, X_{2}, ..., X_{k}) of the tangent space T_{x}(M). Let L_{k}(M) be the set of all k-tangent linear frames u_{k} at all points of M, and let p be the mapping of L_{k}(M) onto M which maps a k-tangent linear frame u_{k} at x into x. The general linear group GL(n;R) acts on L_{k}(M) on the right as follows. If a = (a_{i}^{j}) ∈ GL(n; R) and u_{k} = (X_{1}, X_{2}, ..., X_{k}) is a k-tangent linear frame at x then u.a is, by definition, the k-tangent linear frame (Y_{1}, Y_{2}, ..., Y_{k}) at x defined by

Y_{i} = ∑_{j=1}^{j=k} a_{i}^{j}X_{j}. It is clear that GL(n; R) acts freely on L_{k}(M) and p(u_{k}) = p(v_{k}) iff

v=u.a for some a ∈ GL(n; R). It is known (see [4]) that (L_{k}(M), p, M, GL(n; R)) is a principal fibre bundle and it is denoted by L_{k}(M). We call L_{k}(M) the bundle of k-tangent linear frames over M.

In particular, when k=n, then L_{n}(M) = L(M) is called the bundle of tangent linear frames over M.

The tangent bundle T (M) over M is the bundle (T (M), π, M, R^{n}, GL(n; R)) associated with the bundle of tangent linear frames L(M) over M with the standard fibre R^{n}.

We consider the pair (ξ, G), where ξ = (L_{n}(M), p.M, GL(n; R)) is the principal fibre bundle of tangent linear frames over M and G is the following sequence

G = (GL(n; R) = G_{n} ⊃ G_{n−1} ⊃ ... ⊃ G_{1} ⊃ G_{0} = {e}),

where G_{k} = \{a = (a_{j}^{i}) ∈ G_{n}|a_{j}^{i} = δ_{j}^{i}, j = \overline{1, n − k}, i = \overline{1, n}\}.

It is known that the quotient manifold G_{n}/G_{n−k} is diffeomorphic with the Stiefel manifold V_{n,k} of all systems formed by k linear independent vectors of R^{n}.

We construct the tissue associated to pair (ξ, G). We have [ξ, G] = {ξ_{jk}|0 ≤ j < k ≤ n}, where ξ̃_{jk} = (L_{n}(M)/G_{n−k}, p̃_{jk}, L_{n}(M)/G_{n−j}, G_{n−j}/G_{n−k}, G_{n−j}), since the largest normal subgroup of G_{n} included in G_{m} is G_{0}.

For all 1 ≤ j ≤ n − 1 there exist a diffeomorphism φ_{j} : L_{n}(M)/G_{n−j} ⟶ L_{j}(M) such that φ_{j}∘p̃_{jk} = p_{jk}∘φ_{j}, where p_{jk} : L_{k}(M) ⟶ L_{j}(M) is the canonical projection.

Using the above diffeomorphism, the fibre bundle ξ̃_{jk} can be replaced by the fibre bundle ξ_{jk} = (L_{k}(M), p_{jk}, L_{j}(M), G_{n−j}/G_{n−k}, G_{n−j}).

Hence, the tissue associated to the pair (ξ, G) is [ξ, G] = {ξ_{jk}|0 ≤ j < k ≤ n}. We have ξ_{0n} = ξ and ξ_{jn}(0 < j < n) is a principal fibre bundle with the structure group G_{n−j}.

If 0 ≤ j < k ≤ n, then ξ_{jk} is the bundle associated to ξ_{jn} with the fibre type G_{n−j}/G_{n−k} and G_{n−j} as structure group.

The refinement of ξ defined by G_{n−k} is the following (ξ_{0n} = ξ; ξ_{0,n−k}, ξ_{n−k,n}), where

ξ_{0,n−k} = (L_{n−k}(M), p_{0,n−k}, M, V_{n,n−k}, G_{n})

is the bundle associated to ξ_{on} with the Stiefel manifold V_{n,n−k} as fibre type and G_{n} as structure group;

ξ_{n−k,n} = (L_{n}(M), p_{n−k,n}, L_{n−k}(M), G_{k})

is the principal fibre bundle with G_{k} as the structure group.

Applying the general properties of the tissus associated to principal differentiable fibre bundle, we have the main results:

Let L_{n}(M) be the principal bundle of tangent linear frames over a n-dimensional manifold M. The structure group GL(n; R) of L_{n}(M) can be reduced to the group G_{n−k}, 1 ≤ k < n iff the fibre bundle ξ_{0,n−k} has a cross section.

b) Let ξ = (L_{n}(V_{n}), p, V_{n}, GL(n; R)) be the principal bundle of tangent linear frames to a n-manifold V_{n} and N_{2} = (GL(n; R) ⊃ D(n, R) ⊃ E) a sequence of GL(n; R), where D(n; R) = {(aδ_{i}^{j})|a ∈ R} is the diagonal subgroup and E = {(δ_{i}^{j})}.

Since, D(n; R) is a normal subgroup of GL(n; R), it follows that the refinement of ξ defined by D(n; R), denoted by (ξ_{02}^{‘} = ξ; ξ_{01}^{‘}, ξ_{12}^{‘}), is formed from the following three principal bundles: ξ, ξ_{01}^{‘} = (t(V_{n}), p_{01}, V_{n}, GP (n − 1; R)) is the principal bundle of tangent directions to V_{n} and ξ_{12}^{‘} = (L_{n}(V_{n}), p_{12}, t(V_{n}), D(n; R)), where GP (n − 1; R) is the (n-1)-dimensional real projective group.

2 Čech-de Rham complex of an open cover

In the sequel, we denote by Ω^{∗} the algebra over R generated by dx_{1}, dx_{2}, ..., dx_{n} with the relations

(dx_{j})^{2} = 0; dx_{i}dx_{j} = −dx_{j}dx_{i} for i ≠ j,

where x_{1}, x_{2}, ..., x_{n} are the coordinates on R^{n}.

For any open subset U of R^{n}, the C^{∞} differential q-forms on U are elements of Ω^{q}(U) = {C^{∞} functions on U} ⊗R Ω^{∗}, i.e., if ω ∈ Ω^{q}(U) then ω = ∑ f_{i_{1}i_{2}...i_{q}}dx_{i_{1}}...dx_{i_{q}}, where f_{i_{1}...i_{q}} are C^{∞} functions.

There is a differential operator d : Ω^{q}(U) ⟶ Ω^{q+1}(U) defined as follows :

i) if f ∈ Ω^{0}(U), then df = ∑ \frac{∂f}{∂x_{i}} dx_{i};

ii) if ω =∑ f_{i_{1}...i_{q}}dx_{i_{1}}...dx_{i_{q}}, then dω =∑ df_{i_{1}...i_{q}}dx_{i_{1}}...dx_{i_{q}}.

The complex Ω^{∗}(U) = ⨁_{q=0}^{n} Ω^{q}(U) together with the differential operator d is called

the de Rham complex on U. The kernel of d is called the closed forms and the image of d , the exact forms.

The q-th de Rham cohomology of U is the vector space

H_{DR}^{q}(U) = {closed q − forms}/{exact q − forms}.

We also write H^{q}(U) for q-th de Rham cohomology of U.

Let U be an open cover{U, V } of a manifold M. There is a sequence of inclusions of open sets

M ⟵ U ∐ V ⟵^{∂_{0}} ; ⟵^{∂_{1}} U ∩ V,

where U ∐ V is the disjoint union of U and V and ∂_{0}, ∂_{1} are the inclusions of U ∩ V in V and in U, respectively.

Applying the contravariant functor Ω^{∗}, we get a sequence of restrictions of forms

Ω^{∗}(M) ⟶ Ω^{∗}(U) ⊕ Ω^{∗}(V) ⟶^{∂_{0}^{∗}} ; ⟶^{∂_{1}^{∗}} Ω^{∗}(U ∩ V) ⟶ O,

where by restriction of a form to a submanifold we mean its image under the pullback map induced by the inclusions.

By taking the difference of the last two maps, we obtain the Mayer-Vietoris (short) exact sequence

(1)  O ⟶ Ω^{∗}(M) ⟶ Ω^{∗}(U) ⊕ Ω^{∗}(V) ⟶^{δ} Ω^{∗}(U ∩ V) ⟶ O,

where δ(ω, τ) = τ − ω.

The Mayer-Vietoris (1) gives rise to a long exact sequence in cohomology

(2)  H^{q}(M) ⟶^{r} H^{q}(U) ⊕ H^{q}(V) ⟶^{δ} H^{q}(U ∩ V) ⟶^{d^{∗}} H^{q+1}(M) ⟶ ... ,

where d^{∗} is the coboundary operator given by

(3)  d^{∗}([ω]) = \{ \begin{matrix}[−d(ρ_{V} ω)] & on U \\ [d(ρ_{U}ω)] & on V, \end{matrix}

where (ρ_{U}, ρ_{V}) is a partition of unity subordinate to cover U and [ω] denotes the cohomology class of the form ω.

We observe that the long exact sequence in cohomology allows one to compute in many cases the cohomology of M from the cohomology of the open subsets U and V.

Instead of a cover with two open sets as in the usual Mayer-Vietoris sequence, consider the open cover U ={U_{α}|α ∈ J} of M, where the index set J is a countable ordered set. Denote the pairwise intersections U_{α} ∩ U_{β} by U_{αβ} (when α < β), triple intersections U_{α} ∩ U_{β} ∩ U_{γ} by U_{αβγ}(when α < β < γ), etc.

There is a sequence of inclusions of open sets

M ⟵ ∐ U_{α_{0}} ⟵^{∂_{0}} ; ⟵^{∂_{1}} ∐ U_{α_{0}α_{1}} ⟵^{∂_{0}} ; ⟵^{∂_{1}} ; ⟵^{∂_{2}} ∐ U_{α_{0}α_{1}α_{2}} \begin{matrix}⟵ \\ ... \\ ⟵ \end{matrix} ,

where ∂_{i} is the inclusion which ,,ignores” the i-th open set, for example, ∂_{0} : U_{α_{0}α_{1}α_{2}} ⟶ U_{α_{1}α_{2}}.

This sequence of inclusions of open sets induces a sequence of restrictions of forms

Ω^{∗}(M) ⟶^{r} ΠΩ^{∗}(U_{α_{0}}) ⟶^{δ_{0}} ; ⟶^{δ_{1}} ΠΩ^{∗}(U_{α_{0}α_{1}}) ⟶^{δ_{0}} ; ⟶^{δ_{1}} ; ⟶^{δ_{2}} ΠΩ^{∗}(U_{α_{0}α_{1}α_{2}}) ⟶ ... ,

where δ_{0}, for instance, is induced from the inclusion ∂_{0} : ∐ U_{αβγ} ⟶ ∐ U_{βγ} and therefore is the restriction δ_{0}; ΠΩ^{∗}(U_{βγ}) ⟶ ΠΩ^{∗}(U_{αβγ}).

We define the difference operator δ:ΠΩ^{∗}(U_{α_{0}α_{1}}) ⟶ ΠΩ^{∗}(U_{α_{0}α_{1}α_{2}}) to be the alternating difference δ_{0} − δ_{1} + δ_{2}.

The following sequence

(4) O ⟶ Ω^{∗}(M) ⟶^{r} ΠΩ^{∗}(U_{α_{0}}) ⟶^{δ} ΠΩ^{∗}(U_{α_{0}α_{1}}) ⟶^{δ} ΠΩ^{∗}(U_{α_{0}α_{1}α_{2}}) ⟶^{δ} ...

is exact and it is called the generalized Mayer-Vietoris sequence.

If U={U_{α}|α ∈ J} is an open cover of M, consider the double complex

C^{∗}(U, Ω^{∗}) = ⨁_{p,q≥0} K^{p,q} = ⨁_{p,q≥0} C^{p}(U, Ω^{q}),

where C^{p}(U, Ω^{q}) = ΠΩ^{q}(U_{α_{0}α_{1}...α_{p}}), i.e., K^{p,q} consists of the ,, p- cochains of the cover U with values in the q- forms”.

For example: K^{0,q} = C^{0}(U, Ω^{q}) = ΠΩ^{q}(U_{α_{0}}), K^{1,q} = C^{1}(U, Ω^{q}) = ΠΩ^{q}(U_{α_{0}α_{1}}). The double complex is equipped with the following two differential operators

δ and d, where δ : C^{p}(U, Ω^{q}) ⟶ C^{p+1}(U, Ω^{q}) is the difference operator and d : C^{p}(U, Ω^{q}) ⟶ C^{p}(U, Ω^{q+1}) is the exterior derivative.

We have the following two sequences

(5)  O ⟶ Ω^{q}(M) ⟶^{r} K^{p,q} ⟶^{δ} K^{p+1,q} ⟶ ...

and

(6)  K^{p,0} ⟶^{d} K^{p,1} ⟶^{d} ... ⟶ K^{p,q} ⟶^{d} K^{p,q+1} ⟶ ... .

The double graded complex C^{∗}(U, Ω^{∗}) = ⨁_{p,q≥0} C^{p}(U, Ω^{q}) is called the Čech-de

Rham complex of the cover U of M and an element of the Čech-de Rham complex is called a Čech-de Rham cochain.

Given the doubly graded complex K^{∗,∗} with commuting operators d and δ, one can associate a singly graded complex K^{∗}, where K^{∗} = ⨁_{p+q=n} K^{p,q} and defining the

differential operator D by D = δ + (−1)^{p}.d , on K^{p,q}.

In the sequel we will use the same symbol C^{∗}(U, Ω^{∗}) to denote the double complex and its associated single complex.

The double graded complex C^{∗}(U, Ω^{∗}) computes the de Rham cohomology of M, i.e.

(7)  H_{D}{C^{∗}(U, Ω^{∗})}≅H_{DR}^{∗}(M).

We have

H_{DR}^{n}(M) = ⨁_{p+q=n} H_{D}{C^{∗}(U, Ω^{∗})}.

Let U = {U_{α}|α ∈ J} be a good cover of M (i.e., all finite intersections U_{α_{0}} ∩ U_{α_{1}} ∩ ... ∩ U_{α_{p}} are diffeomorphic to R^{n}) and we denote by H^{∗}(U, R) the Čech cohomology of the cover U.

If U is a good cover of M, then the double complex C^{∗}(U, Ω^{∗}) computes the Čech cohomology of the cover U of M, i.e.

(8)  H^{∗}(U, R)≅H_{D}{C^{∗}(U, Ω^{∗})}.

Therefore, if U is a good cover of the manifold M, then there is an isomorphism between the de Rham cohomology of M and the Čech cohomology of the good cover U of M, i.e.

(9)  H_{DR}^{∗}(M)≅H^{∗}(U, R).

This result provides us with a way of computing the de Rham cohomology by means of combinatorics.

3 Čech-de Rham cohomology of a refinement

Theorem 1. Let [ξ, N_{q}] be a totally trivial tissue (i.e. the bundles ξ_{jq} are product bundles for each 0 ≤ j < q) associated to pair (ξ, N_{q}). Then the following assertions hold

(i)  H_{DR}^{∗}(E_{j})≅H_{DR}^{∗}(B) ⊗ H_{DR}^{∗}(F_{1}^{0}) ⊗ ... ⊗ H_{DR}^{∗}(F_{j}^{j−1})

(ii)  H_{DR}^{∗}(F_{j}^{0})≅H_{DR}^{∗}(F_{1}^{0}) ⊗ H_{DR}^{∗}(F_{2}^{1}) ⊗ ... ⊗ H_{DR}^{∗}(F_{j}^{j−1})

for all 0 ≤ j ≤ q. Proof. The tissue [ξ, N_{q}] being totally trivial it follows that the space E_{j} = E/F_{j} is homeomorphic with B ×F_{1}^{0} ×...×F_{j}^{j−1} and the homogeneous space F_{j}^{0} is homeomorphic with F_{1}^{0} × F_{2}^{1} × ... × F_{j}^{j−1} (see,[5],Th.3.) For j=1, the spaces E_{1} and B × F_{1}^{0} are homeomorphic and H_{DR}^{∗}(E_{1}) = H_{DR}^{∗}(B × F_{1}^{0}). But by Künneth’s formula, we have H_{DR}^{∗}(B × F_{1}^{0})≅H_{DR}^{∗}(B) ⊗ H_{DR}^{∗}(F_{1}^{0}), and we obtain H_{DR}^{∗}(E_{1})≅H_{DR}^{∗}(B) ⊗ H_{DR}^{∗}(F_{1}^{0}).

This means

H_{DR}^{n}(E_{1}) = ⨁_{p+q=n} H_{DR}^{p}(B) ⊗ H_{DR}^{q}(F_{1}^{0}).

Applying now the induction and the general properties of tensor product, by similar arguments we obtain the isomorphisms (i) and (ii).

In the sequel we suppose that the base space B of the principal bundle ξ = (E, p, B, G) has a finite good cover. Theorem 2. Let ξ = (E, p, B, G) be a principal bundle such that B has a finite good cover. If F_{j} (j fixed) is a closed subgroup of G such that the cohomology of G and F_{j} are finite-dimensional, then for the refinement (ξ; ξ_{0j}, ξ_{jq}) of ξ defined by F_{j} the following assertions hold

(i)  H_{DR}^{∗}(E)≅H_{DR}^{∗}(B) ⊗ H_{DR}^{∗}(G)

(ii)  H_{DR}^{∗}(E)≅H_{DR}^{∗}(E_{j}) ⊗ H_{DR}^{∗}(F_{j}).

Proof. (i) The space of cohomology of E (for every n) being a vector space follows that it has a base, that this there are global cohomology classes {e_{i}|i ∈ I} on E. If we restrict {e_{i}} to each fiber of ξ imply that {e_{i}} generate the cohomology of the fiber G, and we can extract a base of H_{DR}^{n}(G), since the cohomology of G is finite-dimensional.Therefore, there are global cohomology classes e_{1}, e_{2}, ..., e_{r} on E which when restrict to each fiber freely generate the cohomology of fiber. Hence,the hypothesis of Leray-Hirsch’s theorem are satisfied for ξ and we have the isomorphisn (i).

(ii) We suppose that U = {U_{i}|i = 1, 2, ..., n} is a finite good cover of B. Then U^{‘} = {p_{oj}^{−1}(U_{i})|i = 1, 2, ..., n} is a finite good cover of E_{j}. Hence the base space E_{j} of the bundle ξ_{jq} has a finite good cover. Using now Leray-Hirsch’s theorem and the fact that the cohomology of F_{j} are finite-dimensional the same argument from proof of (i) gives the isomorphism (ii). Corollary 1. Let ξ = (E, p, B, G) be a principal bundle such that the base space B and the structure group G are compact spaces. Then for the refinement (ξ; ξ_{0j}, ξ_{jq}) of ξ defined by a closed subgroup F_{j} of G, the following assertions hold

(i)  H_{DR}^{∗}(E)≅H_{DR}^{∗}(B) ⊗ H_{DR}^{∗}(G)

(ii)  H_{DR}^{∗}(E)≅H_{DR}^{∗}(E_{j}) ⊗ H_{DR}^{∗}(F_{j}).

Proof. The base space B of ξ has a finite good cover since the manifold B is compact. The hypothesis of Theorem 2 are verified since the cohomology of a compact manifold is finite-dimensional. Applying now Theorem 2 we obtain the isomorphisms (i) and (ii). Theorem 3. Let ξ = (E, p, B, G) be a principal bundle such that the base space B has a finite good cover. Let F_{j} (j- fixed) be a closed subgroup of G such that the cohomology of F_{j} and F_{j}^{0} are finite-dimensional. Then for the refinement (ξ; ξ_{0j}, ξ_{jq}) of ξ defined by F_{j} the following assertions hold

(i)  H_{DR}^{∗}(E)≅H_{DR}^{∗}(E_{j}) ⊗ H_{DR}^{∗}(F_{j})

(ii)  H_{DR}^{∗}(E_{j})≅H_{DR}^{∗}(B) ⊗ H_{DR}^{∗}(F_{j}^{0})

(iii)  H_{DR}^{∗}(E)≅H_{DR}^{∗}(B) ⊗ H_{DR}^{∗}(F_{j}^{0}) ⊗ H_{DR}^{∗}(F_{j}).

Proof. (i) We have that the base space E_{j} of ξ_{jq} has a finite good cover and the cohomology of the fibre F_{j} is finite-dimensional. Applying now Leray-Hirsch’s theorem,we obtain the isomorphism (i).

(ii) We have that the base space B of ξ_{0j} has a finite good cover and the cohomology of the fibre F_{j}^{0} is finite-dimensional. We can apply Leray-Hirsch’s theorem and we obtain the isomorphism (ii).

(iii) This isomorphism results from (i) and (ii). Theorem 4. Let ξ = (E, p, B, G) be a principal bundle such that the base space B has a finite good cover and the structure group G is a connected Lie group. If F is a maximal compact subgroup of G, then for the refinement (ξ; ξ_{01}, ξ_{12}) of ξ defined by F the following assertions hold

(i)  H_{DR}^{∗}(E)≅H_{DR}^{∗}(E/F) ⊗ H_{DR}^{∗}(F)
(ii)  H_{DR}^{∗}(E/F)≅H_{DR}^{∗}(B) ⊗ H_{DR}^{∗}(G/F)
(iii)  H_{DR}^{∗}(G)≅H_{DR}^{∗}(F) ⊗ H_{DR}^{∗}(G/F)
(iv)  H_{DR}^{∗}(E)≅H_{DR}^{∗}(B) ⊗ H_{DR}^{∗}(G/F) ⊗ H_{DR}^{∗}(F).

Proof. Since F is a maximal compact subgroup of G imply, by Ivasawa’s theorem, that G is homeomorphic with the direct product of F and a Euclidian space ( i.e., G is homeomorphic to F × R^{m}). Then H_{DR}^{∗}(G) = H_{DR}^{∗}(F × (G/F)) and using the Küunneth’s formula it follows the isomorphism (iii). Since F is compact and G/F is a Euclidean space it follows that the cohomology of F and G/F are finite-dimensional. Hence the hypothesis of Theorem 3 are verified and we obtain the isomorphisms (i) , (ii) and (iv). Theorem 5. Let ξ = (E, p, B, G) be a principal bundle such that the base space has a finite good cover and the structure group is a simply connected Lie group. Let F be a normal closed subgroup of G such that the factor group G/F is abelian. If the cohomology of G is finite-dimensional, then for the refinement (ξ; ξ_{01}, ξ_{12}) of ξ defined by F the following assertions hold:

(i)  H_{DR}^{∗}(E)≅H_{DR}^{∗}(B) ⊗ H_{DR}^{∗}(G)
(ii)  H_{DR}^{∗}(E/F)≅H_{DR}^{∗}(B) ⊗ H_{DR}^{∗}(G/F).

Proof. (i) We apply the same argument used in the proof of Theorem 1. (i).

(ii) Since F is a normal closed subgroup of G follows that ξ_{01} is a principal bundle having G/F as structure group. But G/F being a simply connected Lie group imply that there is an integer m such that G/F is diffeomorphic with the Euclidian space R^{m}. Hence, ξ_{01} = (E/F, p_{01}, B, G/F) is a principal bundle for which the fibre is diffeomorphic with a Euclidean space. Then there exists a cross section of ξ_{01} defined on B. Applying Theorem 1 from [7], p.36, it follows that ξ_{01} is a trivial bundle; hence, E/F and B×(G/F) are diffeomorphic.We have that H_{DR}^{∗}(E/F) = H_{DR}^{∗}(B×(G/F)). Using now the Künneth’s formula we obtain (ii). Example 2. Let (ξ_{02}^{‘} = ξ; ξ_{01}^{‘}, ξ_{12}^{‘}) the refinement of

ξ = (L_{n}(V_{n}), p, V_{n}, GL(n, R))

defined by F = GL(n, R), see Example 1 (b). If the base B of ξ has a finite good cover or is a compact space, then:

H_{DR}^{∗}(L_{n}(V_{n}))≅H_{DR}^{∗}(V_{n}) ⊗ H_{DR}^{∗}(GL(n, R))
H_{DR}^{∗}(L_{n}(V_{n})≅H_{DR}^{∗}(t(V_{n})) ⊗ H_{DR}^{∗}(R^{m}).

Acknowledgements. A version of this paper was presented at the First Conference of Balkan Society of Geometers, Politehnica University of Bucharest, September 23-27, 1996.

References

[1] A.C.Albu and D.Opriş, Holonomy groups of continuous connections. Finsler continuous connections. Tensor N.S., 39,1962, 105-112.

[2] I.D.Albu, Contribuţii la teoria varietăţilor diferenţiabile paralelizabile, Univ. Timişoara, teză de doctorat, 1980.

[3] R.Bott and L.W.Tu, Differential forms in algebraic topology, Springer-Verlag, New-York, 1982.

[4] S.Kobayashi and K. Nomizu, Foundations of differential geometry, Interscience Publ. Vol. I., 1963.

[5] D.I.Papuc, Sur les raffinements d’un espace fibré principal différentiable, An. St. Univ.“Al. I. Cuza“ Iaşi, Sect. a I-a, Mat., 18(2), 1972, 367-387.

[6] D.I.Papuc, The holonomy group and the refinements of a principal Steenrod bundle, J. Differential Geometry, 11, 1976, 15-22.

[7] N.Steenrod, The topology of fibre bundles, Princeton, 1951.

West University of Timişoara Department of Mathematics Bd. V.Pârvan no 4 1900, Timişoara Romania