Spin-structures and 2-fold coverings ∗

We prove that the existence of a Spin-structure on an oriented real vector bundle and the number of them can be obtained in terms of 2-fold coverings of the total space of the SO(n)-principal bundle associated to the vector bundle. Basically we use theory of covering spaces. We give a few elementary applications making clear that the Spin-bundle associated to a Spin-structure is not sufficient to classify such structure, as pointed out by [6].


Introduction
Let ξ be an oriented n-real vector bundle over a CW-complex X.So ξ has structural group SO(n).There is a classical definition of a Spin-structure of ξ which is given in section 2. This definition is given by two items which are concerning to the existence of a Spin-principal bundle and a 2-fold covering, where some relations hold.The main purpose of this note is to give a proof that the above definition is equivalent just to the existence of some 2-fold coverings.More precisely, the Spin-structures are certain 2-fold coverings of the total space of the associated SO(n)-principal bundle, SO(n) i → P SO(n) (ξ) p −→ X under the usual relation between two covering spaces.Based on this, we give an alternative definition (see Definition 2.3 of a Spin structure).
Using this equivalent definition we easily obtain some known results about Spinstructure, including ones about existence and classification of the Spin-structures.Also it is natural to ask what happens if we look only at the Spin-bundle which arises in the definition of a Spin-structure.Namely, if for a given Spin-structure we consider only the Spin-bundle associated to it, is this sufficient to classify the Spin-structure?We make several calculations which illustrate that this is not the case.The principal bundle maps are an essential part of the structure.
The group SO(n) has fundamental group Z 2 for n > 2 and Z for n = 2.For n > 1, let Spin(n) be the group which is the connected 2-fold covering of the group SO(n).We denote that covering by For the case where n = 1, we define a Spin-structure of an oriented n-real vector bundle ξ to be simply a 2-fold covering of the basis X.Since there is always a 2-fold covering of X, for example X × Z 2 , there is always a Spin-structure.In this case, we define two Spin-structures to be equivalent if the correspondent 2fold coverings are equivalent as covering spaces (see [5], Chapter V section 6).So the study of Spin-structures of an oriented 1-real vector bundle ξ over a space X corresponds to the classical study of 2-fold coverings of X.Also, recall that there is only one orientable 1-real vector bundle ξ over a space X which is the trivial bundle.
From now on let n > 1 and let us assume that the covering spaces are connected.This note contains two sections besides this one.In section 2 we state and prove the main result which is Theorem 2.1.Then we give an alternative definition, Definition 2.3, of a Spin-structure, and we show few results using this new definition.In section 3 we compute the set of Spin-structures in several examples and we look at the set of the Spin-bundles obtained from the Spin-structures.
Similar results as the ones in this note, were obtained by M. Schulz in his Phd.thesis [8].

Statement of the main theorem
2.1.Definitions and Theorem.Let ξ be an oriented n-real vector bundle over a CW-complex X.So ξ has structural group SO(n).Consider the associated is the space of all oriented orthonormal frames.
Recall (see [4] p.371-372) that a G-principal bundle G → P p → X, where G is a group, is given by an atlas where µ G is the product in G and k mn are continuous functions U m ∩ U n → G.They verify the cocycle conditions: and from the definition one can define a right action of G on the total space of the bundle, which commutes with the projection and we denote it by m : P × G → P .We recall other definitions which are going to be used.

Definition 2.1 Given two principal bundles
, 2 a principal bundle homomorphism is a pair (f, λ) where f : P 1 → P 2 is a continuous map and λ : G 1 → G 2 is a group homomorphism such that the following diagram is commutative: where m i is the action of G i on P i .
The group SO(n) has fundamental group Z 2 for n > 2 and Z for n = 2.For n > 1, let Spin(n) be the group which is the connected 2-fold covering of the group SO(n).We denote that covering by Z 2 → Spin(n) λ −→ SO(n) where λ is a group homomorphism.As a result of our discussion in the introduction we will consider always n > 1 and assume that the covering spaces are connected.Definition 2.2 [6,4] Let ξ be an oriented n-real vector bundle over a CW-complex where λ :

map of principal bundles (see Definition 2.1).
2-The Spin-structures (η, f ) and (η , f ) are equivalent if there exists an isomorphism ψ : Given a differentiable manifold which is oriented, we have the notion of Spinmanifold.For, consider the tangent bundle of the manifold which is a SO(n)-bundle as result of the given orientation.So we can apply the Defintion 2.2.When there is a Spin-structure we say that the oriented manifold admits a Spin-structure or it is Spinable.
We recall some constructions of connected covering.
Conversely, let ϕ : π 1 N → Z 2 be an epimorphism.If N denotes the universal cover of N , then the projection p : M = N / ker ϕ → N is a 2-fold covering such that p (π 1 M ) = ker ϕ and the following diagram is commutative (we identify ker ϕ and Z 2 with subsets of the corresponding sets): admit only one epimorphism to Z 2 and, hence, there is a unique (up to covering-equivalence) connected 2-fold covering of SO (2) and SO(n) n ≥ 3, resp.. Now we state the main theorem, which gives an alternative definition of the existence of a Spin-structure.
Let f : E = P / ker ϕ → P = P SO(n) (ξ) be a 2-fold covering and consider the homotopy exact sequence: 1 An oriented n-real vector bundle ξ admits a Spin-structure if and only if there exists a 2-fold covering f : principal bundle of the oriented vector bundle ξ.Further, the set of equivalence classes of 2-fold coverings (as defined by means of diagram 3) as above is in one-to-one correspondence with the set of equivalent classes of Spin-structures (as in Definition 2.2) of the oriented bundle.
Based on the Theorem above we can give the following alternative definition of a Spin-structure on ξ: Definition 2.3 Let ξ be an oriented n-real vector bundle over a CW-complex X.
Now we derive some results using this definition.
Remark 2.1 Given an orientable bundle, one can choose an orientation.If the base X is connected then there are two possible orientations.In any case if ξ is an orientable bundle and ξ 1 is an oriented bundle obtained from ξ by giving an orientation, we can ask for the number of Spin-structures (possibly zero) of this oriented bundle ξ 1 .It is not difficult to see that the number of Spin-structures for ξ 1 is independent of the choice of the orientation of the bundle ξ.In particular there is a Spin-structure of the bundle with respect to one orientation if and only if there is a Spin-structure with another orientation.

Classical results.
As before, let P = P SO(n) (ξ) and define

Corollary 2.1A The cardinality of the set S(ξ) of the Spin−structures on ξ equals the cardinality of A. The set A is either empty or
Proof: The exact sequence on homotopy of the fibration SO(n because When A is not empty then ĩ is an epimorphism.Hence, Hom(π 1 P, Z 2 ) is decomposed into the two cosets modulo kerĩ = Hom(π 1 X, Z 2 ).The non-trivial coset is A. 2 In fact, the exact sequence ( 4) is part of a longer sequence.This longer sequence is obtained as follows: consider the Serre spectral sequence of the fibration where the image of the generator of H 1 (SO(n); Z 2 ) = Z 2 by the morphism w is the second Stiefel-Whitney class of ξ (see [4]).This exact sequence can be rewritten under an equivalent form: Using this sequence together with the previous Corollary we obtain the following well known result: Corollary 2.1B Let ξ be an oriented n-vector bundle over a CW complex X.For the converse, consider ϕ ∈ A and f : E = P /kerϕ → P .We will first show that p • f : E → X is a locally trivial bundle with fiber Spin(n) and then that it is in fact a principal bundle.
By hypothesis SO(n) → P p −→ X is a SO(n)-principal bundle.Let us denote its atlas by (U m , k m ) where {U m } is an open covering of X and k m : p −1 (U m ) → U m × SO(n) a trivialization of the principal bundle.The injection j : V m = p −1 (U m ) → P denotes the injection as a subset of P .The restriction f = f | f −1 (Vm) is a 2-fold covering of V m .By [5] Proposition 11.1 p. 177, we have the equality: As the open sets U m in the atlas (U m , k m ) can be taken contractible, the homeomorphisms k m induce isomorphisms in the fundamental group of π 1 (V m ) and π 1 (SO(n)).Then, the hypothesis ϕ • i being surjective implies that ϕ • j is also surjective, so some H, which is a non-trivial 2-fold covering of U m × SO(n) inducing the identity on U m .So H is a non-trivial 2-fold covering of SO(n), which is unique.This proves that H = Spin(n) and that there exists a homeomorphism h m : (p

See the diagram below:
This means that p • f : E → X is a locally trivial bundle with fiber Spin(n).It remains to show that it is a principal bundle.
By hypothesis where µ SO(n) is the product in SO(n).By construction, there was defined continuous maps The homomorphism λ is surjective.There exists 2 preimages in Spin(n) of k mn (x) ∈ SO(n), denoted by v mnx i , i = 1, 2, such that: In particular for v = e the neutral element of Spin(n) One v mnxi is equal to h mn (x)(e), then where t mnxv is the action of Z 2 .Because t mnxe = 1 and it is continuous in v, it is constant and t mnxv = 1.This proves that the action of h mn (x) on Spin(n) is by 3. What about the Spin−principal bundle which is given in a Spin−structure as defined in Definition 2.2?
Recall that in the Definition 2.2 a Spin-structure is a pair (η, f ) where η is a Spin-principal bundle.In [6], Milnor pointed out that there may exist only one Spin(n)-principal bundle over X, up to bundle equivalence, but different Spinstructures on ξ, where ξ is an oriented bundle over X.
A slightly more general situation can be described as follows.We can construct a map which associates to each Spin-structure (η, f ) the Spin-principal bundle η.It is natural to ask if the Spin-structure can be distinguished by its Spin-principal bundle.In this section we compute the set of Spin-structures as well the set of all Spin-principal bundles obtained from the Spin-structures.In some cases the examples show that the answer of the question above is "yes" and in the other cases the answer is "no".The examples where the answer is "no" illustrate precisely the situation pointed out by Milnor [6].
Our first example is an orientable bundle of dimension 2 over S 1 .The subgroups of index 2 of π 1 P SO(2) (ξ) = Z × Z are the kernels of surjective homomorphisms of Z × Z to Z 2 .There are three surjective homomorphisms: Then The universal cover of S 1 × S 1 is: The operation of H i on R × R is the restriction of the operation of Z × Z on R × R. We denote by E i = (R × R)/H i the 2-fold covering of S 1 × S 1 with fundamental group H i .The projection f i : Now we have to select the double coverings which provide the Spin-structures.c) Eliminate one of the coverings of Remark that (ϑ, µ) and (ϑ + 2π, µ) are not in the same class mod H 1 .As usual, it is possible to define As usual, it is possible to define is well defined.We remark that (ϑ, µ) and (ϑ, µ + 2π) are not in the same class mod H 3 but (ϑ, µ + 2π) and (ϑ + 2π, µ) are in the same class mod H 3 ; hence In the sense of Definition 2.2, the two coverings E 1 and E 3 define different Spin-structures on ξ.
It is worth to mention that the Z 2 -coverings f 1 and f 3 are equivalent to the bounding Spin-structure on S 1 and to the Lie group Spin-structure on S 1 , respectively, as defined by Kirby in [3] pg. 35 and 36.
We found out that only the homomorphisms ϕ 1 • i and ϕ 3 • i are surjective.From Theorem 2.1, this property implies that (f i , λ) is a principal bundle map.The Spin(2)-principal bundles associated to the Spin-structures E 1 , E 3 are orientable S 1 -bundles, so they are classified by homotopy classes of maps S 1 → BSpin(2) = BS 1 = CP ∞ .Since CP ∞ is simply connected, there is only one homotopy class of maps S 1 → BS 1 = CP ∞ .This homotopy class represents the trivial Spin(2)principal bundle.So we conclude that the Spin-bundles associated to the two different Spin-structures are isomorphic.This gives an example of the phenomenon pointed out by Milnor in [6].
Remark 3.1 Since Spin(n)-principal bundles over a space X are classified by the set of homotopy classes of maps [X, BSpin(n)], the above example shows that in general one can not expect that the set of Spin-structures can be identified with the set of homotopy classes of maps [X, BSpin(n)].Furthermore, the example above has the property that any map X → BSpin(n) is a homotopy lifting of a classifying map ϕ ξ : X → BSO(n) of the given bundle ξ, through the map BSpin(n) → BSO(n).Hence, this shows that even if you consider the set of maps X → BSpin(n) which are homotopy liftings of a classifying map ϕ ξ : X → BSO(n) of the given bundle ξ, through the map BSpin(n) → BSO(n), the set of homotopy classes of such maps will not classify the Spin-structures.Remark 3.2 Although the Spin-bundles do not classify the Spin-structures, as shown by the example above, following [3] p. 34 we have the following alternative description of the Spin-structures in terms of homotopy classes of maps.Given a SO(n)-principal bundle let f : X → BSO(n) a map which classsify the bundle.Consider the set L of all maps f : X → BSpin(n) which are liftings of f with respect to the map Bλ : BSpin(n) → BSO(n).There is a one-to-one correspondence