VECTOR BUNDLES OVER THREE-DIMENSIONAL SPHERICAL SPACE FORMS

In this work we consider the class of the compact connected three-dimensional manifolds with positive constant curvature, also known as the three-dimensional spherical space forms. These spaces, or subclasses like generalized quaternions or lens spaces, appear in many different contexts in topology and geometry, and have been completely classified; it is thus natural to ask if we can also count the bundles over them. We answer positively to this question, and give tables in Section 5 to describe all the vector bundles of rank less than 3 over any three-dimensional spherical space form. Besides, in Section 2, we show that, under reasonably wide assumptions on the structure group G, G-bundles over any low (lower or equal to three)-dimensional manifolds can be counted effectively.


Introduction
In this work we consider the class of the compact connected three-dimensional manifolds with positive constant curvature, also known as the three-dimensional spherical space forms.These spaces, or subclasses like generalized quaternions or lens spaces, appear in many different contexts in topology and geometry, and have been completely classified; it is thus natural to ask if we can also count the bundles over them.We answer positively to this question, and give tables in Section 5 to describe all the vector bundles of rank less than 3 over any three-dimensional spherical space form.Besides, in Section 2, we show that, under reasonably wide assumptions on the structure group G, G-bundles over any low (lower or equal to three)-dimensional manifolds can be counted effectively.

Bundles over low-dimensional manifolds
Let G be a Lie group and M m a closed manifold of finite dimension m = 1,2, or 3. Let Ꮾ(M,G) be the set of the equivalence classes of principal G bundles over M. Recall that Ꮾ(M,G) = [M,BG] and, by dimensional reason and since π 2 (G) = 0, they coincide with the set [M,(BG) m ], m = 1 or 2, where (BG) m is the space appearing at level m in the Postnikov decomposition of BG.Thus, when G is connected, When G is not connected we need local coefficients.We can proceed as in [7] and use the Larmore spectral sequence [6].We introduce the following quite general assumption: we assume that the projection p 0 : G → G/G 0 to the quotient by the connected component of the identity has a continuous section s : G/G 0 → G.If this is the case, then G = G/G 0 α G 0 , for some homomorphism α : G/G 0 → Aut(G 0 ), and we can apply [7, Proposition 1].
Proposition 2.1.The classifying space BG is the total space of a G/G 0 -bundle over B(G/G 0 ) with fibre BG 0 and projection Bp 0 : BG → B(G/G 0 ).Moreover, the splitting map s induces a cross-section Bs : B(G/G 0 ) → BG.
Hence, the relevant Postnikov sections are twisted Eilenberg-Mac Lane spaces, and we obtain the exact sequence of sets When M has dimension 1, this gives where the action is by conjugation, namely (φ,α)(x) → φ α (x) = α −1 φ(x)α, and PG = G/ZG denotes the quotient by the center.When M has dimension 2 or 3, we need to compute (Bp 0 ) ) for all [u] ∈ [M,B(G/G 0 )] 0 , and as before we can enumerate the elements [u] ∈ [M,B(G/G 0 )] 0 by the correspondent elements φ u ∈ Hom(π 1 (M),π 0 (G)).We can use the Larmore spectral sequence [6] as in [7] (that has trivial differential in this case).We obtain (Bp 0 ) −1 * (u φ ) = H 2 (M;π 1 (G,u φ )), and hence Eventually, we need to get the quotient by the action of π 0 (G) to get free classes.It follows from [6, Theorem 2.2.2] that the operation + commutes with the action of π 0 (G) as follows: , and hence the quotient can be taken on the group of the homomorphisms.In summary, we have proved the following theorem.
Theorem 2.2.Let G be a compact Lie group satisfying the above assumption and M a closed manifold of dimension 1, 2, or 3.Then, (2.5) Esdras Teixeira Costa et al. 3 Notice that the action of π 0 (G) is trivial whenever π 0 (G) is abelian (in particular if G is abelian) and BG is 2-simple.

Twisted cohomology of 3-dimensional spherical space forms
Let p : F → F/R = G be a presentation for a finite group G, where F and R are free on the sets S and T, respectively.By [3] or [1] we obtain a free resolution of Z over ZG as follows.Let A and B denote ordered sets of abstract module's generators, one generator for each element in the corresponding set of the group's generators S and T, respectively, let e be a single abstract generator, and define the homomorphisms where s b and r a denote the elements in the group's generators set corresponding to the abstract basis elements, and we recall that the group derivation is defined on the elements of F by d s 1 = 0, d s (uv) = d s (u) + ud s v, and d si s j = δ i j , for all s ∈ S. A free resolution of Z over G is then Let Γ be a finite subgroup of SO 4 (R) operating freely on the three sphere S 3 .The quotient spaces S Γ = S 3 /Γ are three-dimensional Riemannian orientable closed manifolds called (orthogonal) spherical space forms [10].A first complete classification of these manifolds was given implicitly by Hopf [5] and in more details by Seifert and Threlfall [8].This classification is given by the list of the possible groups Γ (see also [4]).They are (for presentations see Section 4) (1) the cyclic group C(n), the generalized quaternionic group Q(4n), the binary tetrahedral group T * (24), the binary octahedral group O * (48), and the binary icosahedral group I * (120); (2) the semidirect products (4) the product of any of the above groups with a cyclic group of coprime order.Since S Γ is the three-skeleton of the Eilenberg Mac Lane space K(1,Γ), and all the groups appearing in the above list are finite and finitely presented, the ZΓ-chain complex for the universal covering space S Γ ( ∼ = S 3 ) is given by the resolution (3.2).This provides the chain complex only up to level 2, but this is enough for our purpose since we can dualize the complex to compute the first cohomology groups and eventually apply a generalized version of the Poincaré duality, that holds here without restrictions since the manifolds are orientable, to complete the calculations.

Calculations
In this section we do the necessary calculations in order to apply Theorem 2.2 for the real vector bundles over the spherical space forms of dimension 3. Thus, M = S Γ (Γ being one of the groups listed in Section 3), Notice that G is abelian and BG is 2-simple in all cases except one, when the action of π 1 (BG) corresponds to a change in the local orientation of the bundle.Actually, this case never arises, as appears from the tables in Section 5.
We proceed with the calculations as follows.Each time, we first compute Hom(π 1 (S Γ ), π 0 (G)) = Hom(Ab(Γ),Z/2), that corresponds to the set of the real line bundles over S Γ .Next, we need the cohomology of S Γ , twisted by all these line bundles if n = 2. Let u be an element in [S Γ ,B(G/G 0 )] that classifies a line bundle, and let φ u in Hom(π 1 (S Γ ),π 0 (G)) be the corresponding homomorphism.We need to compute H 2 (S Γ ;π 1 (G,u)).When G = O 2 , since (BG) 1 = G/G 0 , the sheaf π 1 (G,u) with fibre π 1 (G) = Z and group π 0 (G) = Z/2, acting by the automorphism determined by a representation ρ : π 0 (G) → Aut(π 1 (G)), corresponds bijectively to (is classified by) the twisting homomorphisms φ : π 1 (S Γ ) → π 0 (G), that is, we can identify (in the other cases we just need the trivial sheaf) Therefore, for each Γ and each allowed representation for it, we give the explicit form of the chain complex described in Section 3, and we compute the twisted homology groups.We will use the following notation for the groups representations: for a subset W of the set of the generators of Γ, let ρ W : Γ → Aut(Z) denote the homomorphism determined by ρ W (W) = −1 and ρ W (S\W) = 1; ρ 0 will denote the trivial representation.Observe that not all such W define a homomorphism, the relations of the presentation of Γ impose restrictions on that.Notice also that, whenever we know a complete chain complex, we write it down explicitly.In this case we have a full periodic resolution, see [3] or [2], that gives the chain complex The homology groups in the above representations are Twisted homology Esdras Teixeira Costa et al. 5 while the cohomology with global Z/2 coefficients is where (n,m) denotes the gcd of n and m. (4.

Vector bundles over 3-dimensional spherical space forms
In this section we give a complete enumeration of the real vector bundles of ranks 1,2, and 3 over the 3-dimensional spherical space forms.The enumeration is given in Tables 5.1 and 5.2, where we use the following notation.In Table 5.1, for each group Γ, we list in the first column the line bundles that are counted by their Stiefel-Whitney class; thus, 1 denotes the trivial bundle.Here, the α s are fixed generators of H 1 (S Γ ;Z/2)-note that we can identify this set with Hom(Γ,Z/2).In the second column are listed, for each line bundle with first SW class α s , the 2 bundles with the same first SW class.These 2 bundles are counted by expressions like 1 + α s + y, where y is the obstruction class in H 2 (S Γ ;Z ρs ).
10 Vector bundles over three-dimensional spherical space forms Here, the β i are fixed generators of H 2 (S Γ ;Z ρs ).In Table 5.2 appear the real vector bundles of rank 3 with the same notation (but the β i are generators of H 2 (S Γ ;Z/2)).