A counting invariant for maps into spheres and for zero loci of sections of vector bundles

The set of unrestricted homotopy classes $[M,S^n]$ where $M$ is a closed and connected spin $(n+1)$-manifold is called the $n$-th cohomotopy group $\pi^n(M)$ of $M$. Moreover it is known that $\pi^n(M) = H^n(M;\mathbb Z) \oplus \mathbb Z_2$ by methods from homotopy theory. We will provide a geometrical description of the $\mathbb Z_2$ part in $\pi^n(M)$ analogous to Pontryagin's computation of the stable homotopy group $\pi_{n+1}(S^n)$. This $\mathbb Z_2$ number can be computed by counting embedded circles in $M$ with a certain framing of their normal bundle. This is a analogous result to the mod $2$ degree theorem for maps $M \to S^{n+1}$. Finally we will observe that the zero locus of a section in an oriented rank $n$ vector bundle $E \to M$ defines an element in $\pi^n(M)$ and it turns out that the $\mathbb Z_2$ part is an invariant of the isomorphism class of $E$. At the end we show, that if the Euler class of $E$ vanishes this $\mathbb Z_2$ invariant is the final obstruction to the existence of a nowhere vanishing section.


Introduction
Pontryagin computed in [14] the (stable) homotopy group π n+1 (S n ) (n ≥ 3) by using differential topology. Let us describe briefly his construction, since this paper will generalize his idea.
Pontryagin showed that π n+1 (S n ) is isomorphic to the bordism group of closed 1-dimensional submanifolds of R n+1 furnished with a framing on its normal bundle (a framing is a homotopy class of trivializations, see section 2). We denote this bordism group by Ω fr 1 (R n+1 ). Let (C, ϕ) be a representative of an element of Ω fr 1 (R n+1 ), i.e. C is a union of embedded circles in R n+1 and there are maps ϕ 1 , . . . , ϕ n : C → R n+1 such that (ϕ 1 (x), . . . , ϕ n (x)) is a basis of ν(C) x for every x ∈ C. Let ϕ n+1 be a trivialization of the tangent bundle of C. Then (ϕ 1 (x), . . . , ϕ n+1 (x)) is a basis of R n+1 for every x ∈ C. Without loss of generality we may assume that ϕ 1 , . . . , ϕ n+1 is pointwise an orthonormal basis. If (e 1 , . . . , e n+1 ) denotes the standard basis of R n+1 then consider the map A = (a ij ) : C → SO(n + 1) such that ϕ i (x) = n+1 j=1 a ij (x)e j for x ∈ C. Let π 1 (SO(n + 1)) be identified with Z 2 , then Pontryagin defines [14,Theorem 20] δ(C, ϕ) := [A] + (n(C) mod 2) where [A] denotes the homotopy class of A in π 1 (SO(n + 1)) and n(C) is the number of connected components of S. He showed that δ is well-defined on Ω fr 1 (R n+1 ) and is an isomorphism of groups. From a different point of view, one may consider his computation not as a computation of a homotopy group of S n but rather of a cohomotopy group of S n+1 . If X is a CW space then the cohomotopy set of X is defined as the set of (unrestricted) homotopy classes π n (X) := [X, S n ], cf. [3,15]. The set π n (X) for X a finite CW complex of dimension n + 1 carries naturally a group structure, which is described in the beginning of section 4. Steenrod showed [16, Theorem 28.1, p. 318] that π n (X) fits into a short exact sequence 0 −→ H n+1 (X; Z 2 )/Sq 2 µ(H n−1 (X; Z)) −→ π n (X) −→ H n (X; Z) −→ 0, where µ : H * (X; Z) → H * (X; Z 2 ) is the mod 2 reduction homomorphism. Here the surjective map is the Hurewicz homomorphism which assigns to every f ∈ π n (X) the cohomology class f * (σ) ∈ H n (X; Z) where σ ∈ H n (S n , Z) is a fixed generator.
If X = M is a manifold then the second Wu class [21] is equal to the second Stiefel-Whitney class w 2 (M ), hence Sq 2 (x) = w 2 (M ) ⌣ x for x ∈ H n−1 (M ; Z 2 ). Therefore if M is spin then π n (M ) fits into the exact sequence as abelian groups. However the splitting map is constructed in a purely homotopy theoretic setting and an aim of this article is to provide a geometric description in case M is a spin manifold. This splitting map κ : π n (M ) → Z 2 (see Definition 3.8) for (ST) will be constructed similarly to Pontryagin's invariant δ from above. An important ingredient in Pontryagin's construction was the canonical background framing by the standard basis of R n+1 , which allowed him to define the map A : S → SO(n + 1). In general if we replace S n+1 or R n+1 by M , this background framing is not available any more. But this can be circumvented by using the spin structure of M , since over a circle every vector bundle with a spin structure defines a certain framing, cf. Lemma 3.1. Section 4 is devoted to determine geometrically the kernel of the Hurewicz map π n (M ) → H n (M ; Z). Finally we show that the splitting map possesses a naturality property, cf. Proposition 4.3 and that for a map f : M → S n the number κ(f ) can be described by a counting formula, cf. Corollary 4.4. This is an analogous result to the mod 2 Hopf theorem, see [13,4]. It should be mentioned that in [7] the authors discuss the case n = 3 and in [9] a similar construction of a Z 2 invariant was used to classify quaternionic line bundles over closed spin 5-manifolds.
In Section 5 we will apply the results of Sections 3 and 4 to the theory of vector bundles. Suppose E → M is a oriented vector bundle of rank n over a closed spin (n + 1)-manifold M . Then any section of E which is transverse to the zero section defines by means of its zero locus an element of Ω fr 1 (M ) and this element is independent of the transverse section. Thus using κ one defines an invariant κ(E) ∈ Ω fr 1 of the isomorphism class of the bundle E → M . In Theorem 5.5 it is shown, that κ(E) can be regarded as the secondary obstruction to the existence of a nowhere vanishing section. As an application we provide in Example 5.6 a simple proof of the well-known fact, that the maximal number of linear independent vector fields on S 4k+1 is equal to 1. Finally we show that π n (M ) can be mapped injectively into the set of isomorphism classes of oriented rank n vector bundles over spin (n + 1)-manifolds for n = 4 and n = 8, cf. Proposition 5.8.
To be framed bordant is an equivalence relation and the set of equivalence classes is called the bordism classes of normally framed k-dimensional submanifolds denoted by Ω fr k (M ). If (C, ϕ) is a normally framed submanifold then we denote by [C, ϕ] its bordism class in Ω fr k (M ). The Pontryagin-Thom map provides a bijection between π n+1−k (M ) and Ω fr k (M ) as follows (cf. [13,7]): Let f : M → S n+1−k represents an element of π n+1−k (M ). Choose a regular value x 0 ∈ S n+1−k and set C x0 := f −1 (x 0 ). Moreover choosing a basis of the tangent space T x0 S n+1−k endows the normal bundle with a framing ϕ x0 by means of the derivative of f . The bordism class [C x0 , ϕ x0 ] ∈ Ω fr k (M ) is well defined and the map . is a bijection, see [13,Theorem B and A].
A stable framing of a real vector bundle E → C of rank r is an equivalence class of trivializations of E ⊕ ε l for some l ∈ N where two trivializations are considered to be equivalent if there exists some L > l 1 , l 2 such that the isomorphisms are homotopic, cf. [4,Section 8.3]. If E is the tangent bundle of C, then a stable framing of T C is called a stable tangential framing. If E is the normal bundle of an embedding of C into a sphere of big dimension, then we call a stable framing a stable normal framing. We define Ω fr k to be the bordism classes of stably (tangential) framed manifolds. More precisely two stably framed manifolds (C 0 , ϕ 0 ) and (C 1 , ϕ 1 ) where ϕ i : T C i ⊕ ε l → ε k+l is an isomorphism are equivalent if there is a bordism Σ between C 0 and C 1 such that the tangent bundle of Σ possesses a stable framing and the restriction on C 0 and C 1 coincides with the framing ϕ 0 and ϕ 1 respectively. Note that Ω fr k is isomorphic to π S k , the k-stable homotopy group of spheres (cf. [4,Theorem 8.17]) and by the Pontryagin-Thom construction we have Ω fr k = lim − →l Ω fr k (S l ) where we use the equatorial embeddings S l1 ֒→ S l2 if l 1 < l 2 to construct well-defined maps Ω fr k (S l1 ) → Ω fr k (S l2 ). For this article the case k = 1 will be of importance. In this case we have Ω fr 1 ∼ = π S 1 ∼ = Z 2 . Consider a connected and closed 1-dimensional manifold S 0 and stable tangential framing From the discussion above, (S 0 , ϕ 0 ) defines a class in Ω fr 1 and can be realized as follows: Consider . . , n + 1}. Denote e 1 , . . . , e n+1 the canonical basis of R n and E i (x) = e i for x ∈ R n+1 the constant vector fields on R n+1 . Moreover let V (x) = x for x ∈ R n . The normal bundle ν(S 0 ) of S 0 is trivialized by V, E 3 , . . . , E n+1 restricted to S 0 . Using this normal framing we obtain a stable framing where the latter framing is induced by E 1 , . . . , E n+1 . Hence this defines an element in Ω fr 1 (S n+1 ) which represents the framed null bordism, since the framing of ν(S 0 ) can be extended to a properly embedded stably framed disc in S n+1 × [0, 1]. Clearly the non-trivial element of Ω fr 1 (S n+1 ) can be represented by twisting the normal framing E 3 , . . . , E n+1 with a map S 0 → SO(n) such that its homotopy class in π 1 (SO(n)) ∼ = Z 2 is not zero. Every stable tangential framing of a closed and connected 1-dimensional manifold can be obtain in this way.
If E → N is an oriented vector bundle over a manifold N , then we say that E is spinnable if the second Stiefel-Whitney class w 2 (E) is zero. This means that E can carry a spin structure, that is a lift of the classifying map N → BSO(n) to a map N → BSpin(n) in the fibration K(Z 2 , 1) → BSpin(n) → BSO(n). Consequently E is a spin bundle is it spinnable and a spin structure is fixed. If a spin structure is fixed on E → N then any other spin structure is in 1 : 1 correspondence with elements in H 1 (N ; Z 2 ).
We write F (N ) for the frame bundle of a manifold N . If V ⊂ N is a submanifold such that its normal bundle is framed then we obtain an embedding F (V ) ⊂ F (N ). Thus a spin structure on N induces a spin structure on V , cf. [12]. In particular if V is the boundary of a spin manifold N , then V inherits a spin structure from N . Finally if E → N is a vector bundle with a spin structure and V ⊂ N a submanifold, then clearly E| V → V also inherits a spin structure from E → N .
Let E → S 1 be a spinnable vector bundle of rank r ≥ 3 over the unit circle S 1 . Then E has exactly two non-isomorphic spin structures. Clearly E → S 1 can be extended to E → D 2 , where D 2 denotes the closed unit disc in R 2 . Since D 2 is contractible E → D 2 admits a unique spin structure. Restricting this structure to the boundary of D 2 gives a spin structure on E → S 1 , which will be called the standard spin structure. The other should be called the non-standard spin structure. In other words, the standard spin structure on E → S 1 can be extended to D 2 , the non-standard not.

The index of framed circles
We define in this section the key invariant of this article. For its construction the following basic lemma is the crucial observation.
Then E is isomorphic to the trivial bundle and a choice of a spin structure on E determines a framing on E.
Proof. E is isomorphic to the trivial bundles since it is an orientable vector bundle over a circle. Fix a spin structure on E, i.e. let F ′ (E) be a Spin(n)-principal bundle over S 1 which is a two-sheeted cover over the frame bundle F (E) of E. Let π : F ′ (E) → F (E) be the projection which is equivariant with respect to the two-sheeted covering Spin(n) → SO(n). Clearly F ′ (E) is the trivial Spin(n)-principal bundle over S 1 and denote by σ : S 1 → F ′ (E) a global section. Then π • σ is a global section of F (E) hence a trivialization of E → S 1 . Any other such global section σ : S 1 → E differs from σ by a map ϕ : S 1 → Spin(n). Since π 1 (Spin(n)) = 1 the map ϕ has to be null-homotopic which means that the two trivializations π • σ and π • σ have to be homotopic, thus they define the same framing on E.
In the same way one proves Corollary 3.2. Let Σ be a 1-dimensional CW-complex (not necessarily connected) and E → Σ a vector bundle of rank ≥ 3 endowed with a spin structure. Then E is isomorphic to the trivial bundle and the spin structure induces a framing on E. Definition 3.3. Let E → S 1 be a spinnable vector bundle. The framing induced by the standard spin structure on E is called the standard framing and from then non-standard spin structure the non-standard framing.
Example 3.4. The spheres S n+1 admit a unique spin structure which can be constructed as mentioned in the preliminaries, i.e. S n+1 is the boundary of the closed unit ball D n+2 in R n+2 which admits a unique spin structure.
Let S 0 ∈ S n+1 be the intersection of a 2-dimensional linear subspace W ⊂ R n+2 with S n+1 and denote by D 2 0 = W ∩ D n+2 . Thus after Lemma 3.1 T S n+1 | S0 inherits a framing from the spin structure. Denote by ϕ 1 , . . . , ϕ n+1 a trivialization of this framing, then the framing must be null homotopic in SO(n + 2) by the definition of the spin structures of S n+1 and T S n+1 | S0 (such that it lifts to Spin(n + 2)). Thus ϕ must be homotopic the constant framing x → (e 1 , . . . , e n+2 ), where e 1 , . . . , e n+2 denotes the canonical basis of R n+2 . In particular this means, that T S n+1 | S0 inherits the standard framing from the spin structure of S n+1 . Ω fr 1 (M ) possesses a group structure which can be expressed as follows: Having two 1-dimensional closed submanifolds C and C ′ of M which are normally framed, then they are framed bordant in M to framed submanifoldsC andC ′ whose intersection is empty. Taking the equivalence class of the disjoint unionC ∪C ′ with the respective framings yields an abelian group structure on Ω fr 1 (M ), cf. [13, Problem 17 and p. 50].
Next, we construct a homomorphism κ : Ω fr 1 (M ) → Ω fr 1 , where Ω fr 1 is the bordism group of stably framed 1-dimensional closed manifolds. Therefore let (C, ϕ C ) be a closed submanifold of dimension 1, such that its normal bundle ν(C) is framed by ϕ C (thus representing an element in Ω fr 1 (M )). From Lemma 3.1 the bundle T M | C inherits a framing ϕ σ from the spin structure of M . Using also the framing of ϕ C we obtain a stable tangential framing be another normally framed closed 1-dimensional submanifold framed bordant to (C, ϕ C ). Thus there is a bordism Σ ⊂ M × I between C and C ′ such that the normal bundle of Σ in M × I possess a framing ϕ Σ . By definition restricting ϕ Σ to C and C ′ yields ϕ C and ϕ C ′ respectively. Since Σ is homotopy equivalent to a 1-dimensional CW-complex and since M × I inherits a unique spin structure from M we obtain a framing ϕ Σ,σ on T (M × I)| Σ . Of course the framings ϕ Σ,σ restricted to C and C ′ are just the framings ϕ σ and ϕ ′ σ respectively (i.e. induced by the spin structure of T M | C and Remark 3.6. As described above, the group structure of Ω fr 1 (M ) is given by disjoint union of submanifolds and their respective normal framings. Let (C, ϕ) be a framed 1-dimensional closed submanifold of M and denote by C = S 1 ∪ . . . ∪ S k the connected components of C. We may assume that the union is always disjoint. Thus S i is an embedded circle and ϕ i := ϕ| Si a normal framing of S i . Consequently we have Definition 3.7. Let S ⊂ M be an embedded circle and ϕ a framing of ν(S). We call the bordism class [S, ϕ] ∈ Ω fr 1 (M ) a framed circle of M . The corresponding stable class [S, ϕ st ] ∈ Ω fr 1 will be called the the index of [S, ϕ] (with respect to the spin structure of M ) and will be denoted by ind(S, ϕ). Definition 3.8. Let M be an (n + 1)-dimensional closed spin manifold. Then we define a map We call κ the degree map of M with respect to the chosen spin structure. Remark 3.9. It is clear from the construction that κ is a homomorphism.
Let S 0 be the intersection of S n+1 with a 2-dimensional linear subspace W of R n+2 . We argued in Example 3.4 that T S n+1 | S0 inherits the standard framing.
(b) Let N be a closed, simply connected, spin manifold of dimension n. Then M := S 1 × N admits two different spin structures since M is the boundary of D 2 × N which has up to isomorphism a unique spin structure. The two different spin structures on M can be described as follows: One can be extended from M to D 2 × N and the other not. We call the latter one the standard spin structure and the former one the non-standard spin structure of S 1 × N .
For q 0 ∈ N consider the circle S 0 := S 1 × q 0 ⊂ S 1 × N . Clearly we have a canonical isomorphism Thus choosing a basis in T q0 N gives a framing ϕ 0 on ν(S 0 ) which extends to a framing of (D 2 × q 0 ) × T q0 N . Thus we have κ 0 ([S 0 , ϕ 0 ]) = 0 for the standard spin structure and for the non-standard spin structure.
For q 1 ∈ N with q 0 = q 1 we consider C = S 1 × q 0 ∪ S 1 × q 1 with fixed normal framing on S 1 × q i which gives a framing ϕ on C. Then κ([C, ϕ]) is independent of the chosen spin structure of M .
This shows that in general κ will depend on the spin structure. The next proposition will show how it depends from it.
We continue with the description of the " dual" short exact sequence to (ST). There is a natural group homomorphism Ω fr 1 (M ) → Ω SO 1 (M ), which assigns to every framed 1-submanifold [C, ϕ] the oriented bordism class induced by the orientation framing ϕ. This is well-defined since every normally framed bordism in M is also an oriented bordism (M is oriented). By the seminal work of Thom [18] we have an isomorphism Ω SO 1 (M ) → H 1 (M ; Z) which assigns every oriented submanifold its fundamental class in H 1 (M ; Z). Thus we obtain a group homomorphism Φ : Ω fr which is clearly surjective. The kernel of Φ is at most isomorphic to Z 2 and elements of the kernel are represented by framed circles (S, ϕ) such that S is oriented null-bordant, i.e. there is an embedded oriented disc D ⊂ M × I with the properties ∂D = S and the orientations of ∂D and S agree. We may equip the normal bundle of S with two framings. If both framings can be extended over D then the kernel is trivial and otherwise Z 2 .
Lemma 3.12. The restricted degree map κ| ker Φ : ker Φ → Ω fr 1 is an isomorphism. Proof. Since κ is a homomorphism it will map the neutral element of ker Φ to that of Ω fr 1 . Thus it suffices to show the following: Let (S, ϕ) be a framed circle such that S is oriented null-bordant in M but ϕ cannot be extended over the nullbordism. We have to show κ([S, ϕ]) = 0, where 0 denotes the neutral element of Ω fr 1 . We may assume that S lies in a chart of M 1 . Thus we may embed S into R n+1 endowed with a normal framing, which cannot be extended over a nullbordism in R n+1 . Hence the index of (S, ϕ) defines a non-trivial element in Ω fr 1 (note that since w(S) = 0 the element κ[(S, ϕ)] does not depend on the spin structure of M , cf. Lemma 3.11).
Thus we may identify ker Φ with Ω fr 1 via (κ| ker Φ ) −1 and we obtain a short exact sequence and from Lemma 3.12 κ is a splitting map. Therefore We finish this section by giving an alternative way to compute the index of a framed circle in the spirit of Pontryagin [14]. Suppose [S, ϕ] is a framed circle, thus there are trivializations of ν(S) and T M | S such that we obtain the stable framing ε n+1 ∼ = T S ⊕ ε n (where is can assume that the isomorphism is orientation preserving). Denote by v 1 , . . . , v n+1 and by w 2 , . . . , w n+1 the trivializations of T M | S and ν(S) respectively. Let w 1 be a trivialization of T S. Let Φ : T S ⊕ ε n → ε n+1 be the isomorphism of the stable framing, then there is a matrix A = (A ij ) : S → GL + (n + 1) (where GL + (n + 1) is the set of all invertible real matrices of size (n + 1) × (n + 1) with positive determinant) such that such that Since SO(n + 1) is a strong deformation retract of GL + (n + 1) we have π 1 GL + (n + 1) ∼ = Z 2 . The map A : S → GL + (n + 1) defines an element [A] ∈ π 1 GL + (n + 1) . Changing the homotopy classes of trivializations of T M | S and ν(S) does not change [A]. Furthermore [A] is also independent of the choice of trivializations of T S.
According to the Preliminaries in Section 2 any stable framing ind(S, ϕ) can be represented by a framed circle S 0 in R n+1 such that recovers the stable framing of (S, ϕ). It follows that ind(S, ϕ) = δ(S 0 , ϕ 0 ), where δ is the invariant constructed by Pontryagin, [14,Theorem 20]. We will use a different notation: Let us denote by [A] the homotopy class constructed above from the stable framing and by [A] the element [A] + 1 ∈ Ω fr 1 (S n ) ∼ = Z 2 where 1 is the non-trivial element. Thus we proved Lemma 3.14. We identify π 1 GL + (n + 1) with Ω fr 1 by the unique isomorphism Z 2 → Z 2 . Then [A] = ind(S, ϕ).

Computation of π n (M)
We start this section to explain the group structure of π n (M ). Let j : S n ∨ S n → S n × S n be the inclusion of the (2n − 1)-skeleton of S n × S n (endowed with the standard CW structure) then, since M is n + 1-dimensional CW complex, the induced map j # : [M, S n ∨ S n ] → [M, S n × S n ] is an isomorphism. For f, g ∈ π n (M ) the group structure is defined by This makes π n (M ) to an abelian group. Now, let f : M → S n be a differentiable map and x 0 ∈ S n a regular value. We orient S n by the normal vector field pointing outwards and the standard orientation of R n+1 .
Let Ψ : π n (M ) → H n (M ; Z) be the map Ψ([f ]) := f * σ where σ ∈ H n (S n ; Z) is a fixed generator. We define the analogous degree map κ : π n (M ) → π S 1 , where π S 1 is the first stable homotopy group of spheres, as follows: κ is the composition of where the first and the last isomorphism is again induced by the Pontryagin-Thom isomorphism.
Theorem 4.1. Let M be a closed (n + 1)-dimensional spin manifold. Then (a) The generator of ker Ψ ∼ = Z 2 is given by the homotopy class of the map η • ω : M → S n+1 , where η represents a generator of π n+1 (S n ) and ω : M → S n+1 is a map of odd degree. Thus ker Ψ ∼ = π n+1 (S n ).
is an isomorphism of abelian groups.
Proof Since p has odd degree, l has to be an odd number (such maps exists e.g. using the Pontryagin-Thom construction). Furthermore let x 0 ∈ S n be a regular value of η and S 0 = η −1 (x 0 ). We may assume that S 0 is connected (e.g. see [13,Theorem C]) and S 0 ⊂ V . Let ϕ 0 be the framing of ν(S 0 ) induced by η, then 0 = [S 0 , ϕ 0 ] ∈ Ω fr 1 (S n+1 ) ∼ = π n+1 (S n ) ∼ = Z 2 and therefore by definition we have ind(S 0 , ϕ 0 ) = 0. Denote by S i := ω −1 i (S 0 ) and frame ν(S i ) by ϕ 0 and dω i . Then C = S 1 ∪ . . . ∪ S l together with the framings ϕ i is a Pontryagin manifold for η • ω to the regular value x 0 . Note that w(S i ) = 0 for i = 1, . . . , l, since they are contained in a chart of M . By Proposition 3.11 this means that their indices do not depend on the spin structure of M . Clearly we deduce ind(S i , ϕ i ) = ind(S 0 , ϕ 0 ) = 0 for all i = 1, . . . , l and from that we infer since l is odd, which proves (a).
Part (b) follows directly from part (a). Finally we would like to show, that κ is natural with respect to maps between manifolds which preserve the spin structure Proposition 4.3. Suppose Φ : M 1 → M 2 is a map between two closed and connected spin manifolds of dimension (n + 1). We assume that the spin structure of M 1 coincides with the pull-back spin structure by Φ of M 2 . Then for the natural homomorphism Φ # : where deg 2 Φ is the mod 2 degree of Φ. Therefore using the isomorphism Proof. First note that Φ # is well-defined on the homotopy class of Φ. For f ∈ π n (M 2 ) there is a decomposition f = f α + f ν with κ(f α ) = 0, f * α (σ) = α and κ(f ν ) = ν as well as f * ν (σ) = 0. Let us show first Φ # (f α ) = f Φ * (α) . Clearly we have Φ # (f α )(σ) = Φ * (α) thus it remains to show κ(Φ # (f α )) = 0. Let C 2 be the preimage of a regular value of f α with a normal framing ϕ 0 such that κ([C 2 , ϕ 0 ]) = 0. Moreover we may choose f α such that each framed circle of (C 2 , ϕ 0 ) has index 0. Deform Φ to be transversal to C 2 , thus C 1 := Φ −1 (C 2 ) is a closed 1-dimensional submanifold of M 1 . The normal bundle to C 1 is isomorphic to the pull back of the normal bundle of C 2 by Φ. This induces a framing on C 2 such that every framed circle thereof has index 0 (note that the spin structure of M 1 is the pulled back by Φ from M 2 ) which is also the framing induced by the map f α •Φ. But this means κ(Φ # (f α )) = 0.
On the other hand we may assume a preimage of a regular point in S n under f ν is a contractible circle S 2 in M 2 with normal framing ϕ such that the index of the framed circle (S 2 , ϕ) is ν ∈ π n+1 (S n ). Then making again Φ transverse to S 2 we obtain a normally framed submanifold (C 1 , ϕ) such that the index of each framed circle in C 1 has index ν. As in the proof of Theorem 4.1 the degree of (C 1 , ϕ) is just deg 2 Φ · ν. Therefore Φ # (f ν ) = f deg 2 Φ·ν and the proposition follows.

Application to vector bundles
In this section π : E → M should denote an oriented vector bundle of rank n endowed with a spin structure. Let s : M → M be a section. If not otherwise stated, we say s is transversal if s is transversal to the zero section 0 E of E. For a transversal section s the zero locus C is a smooth 1-dimensional closed submanifold of M . The differential ds : T M → T E restricted to ν(C) is an isomorphism of the vector bundles ν(C) → E| C . Since E possess a spin structure, by Lemma 3.1 E| C has a framing and with ds this endows ν(C) with the framing ϕ of E| C . Note that the homology class [C] ∈ H 1 (M ; Z) is the Poincaré dual of the Euler class of E. For the next theorem we will need a technical Lemma. Let D m denote the closed unit ball in R m and consider a smooth map f : D n+k+1 → R n+1 . Assume that 0 ∈ R n+1 is a regular value for f and Σ k f := f −1 (0) does not intersect the boundary of D n+k+1 . Denote by ϕ f the induced framing on ν(Σ k f ). Since Σ k f is a submanifold of R n+k+1 the trivialization ϕ f defines a stable tangential framing of Σ k f thus the pair (Σ k f , ϕ f ) defines an element in Ω fr k . On the other side, consider and choose a regular value y ∈ S n . Denote by (Σ k g , ϕ g ) the induced stably framed manifold.
Lemma 5.4. With the notation above we have that (Σ k f , ϕ f ) and (Σ k g , ϕ g ) are stably framed bordant, thus they define the same element in Ω fr k . Proof. There is an ε > 0 such that the closed ball D ε centered in 0 ∈ R n+1 with radius ε contains only regular values of f . The preimage of D ε under f is a disc bundle D(Σ k f ) of the normal bundle Moreover the Pontryagin manifold (Σ y ′ , ϕ y ′ ) is framed bordant to (Σ f , ϕ f ). Thus we would like to show that (Σ y ′ , ϕ y ′ ) represents the same element in Ω fr k as (Σ g , ϕ g ). Since the normal bundle of S(Σ k f ) is trivial the framing ϕ y ′ induces a framing ϕ ′ y ′ on ν(Σ y ′ ֒→ S(Σ k f )) such that (Σ y ′ , ϕ y ′ ) is stably framed bordant to (Σ y ′ , ϕ ′ y ′ ). But the latter normally framed manifold is the Pontryagin manifold to the map f | S(Σ k f ) : S(Σ k f ) → S ε at the point y ′ ∈ S ε . Let N be the complement of the interior of D(Σ k f ) in D n+k+1 . Then N is a framed cobordism between S n+k = ∂D n+k+1 and S(Σ k f ). The restriction of the map to S n+k is equal to g and F restricted to S(Σ k ) is equal to ε −1f . Hence F defines a framed bordism between (Σ k g , ϕ g ) and (Σ y ′ , ϕ ′ y ′ ) which proves the lemma.
Theorem 5.5. Let E → M be an oriented vector bundle of rank n with w 2 (E) = 0 over a closed spin manifold M of dimension n + 1. Then E admits a nowhere vanishing section if and only if the Euler class is zero and κ(E) = 0.
Proof. Suppose there is a nowhere vanishing section of E then clearly this section is transverse and has an empty framed divisor. Thus from Theorem 3.13 we have that the Euler class must be zero and κ(E) = 0. Assume now that e(E) = 0 and κ(E) = 0. Consider the fibration where BSO(k) denotes the classifying space to the special orthogonal group SO(k). Consider the classifying map g : M → BSO(n) for E → M . There exists a nowhere vanishing section if and only if there is a liftĝ : M → BSO(n − 1) of g up to homotopy. First we put a CW-structure on M (e.g. induced by a Morse function) then over the (n − 2)-skeleton of M there exists such a liftĝ of g. The obstruction to extend the lift over the n-skeleton lies in H n (M ; π n−1 (S n−1 )) = H n (M ; Z) which is given by the Euler class e(E). Since this is assumed to be zeroĝ extends over the n-skeleton of M . The obstruction to extendĝ over the top cell of M lies in H n+1 (M ; π n (S n−1 )) ∼ = π n (S n−1 ) ∼ = Z 2 . Let e n+1 be the top cell of M and ψ : ∂e n+1 ∼ = S n → M the corresponding attaching map. The bundle E| en+1 is canonical isomorphic to e n+1 × R n . Let σ : M → E be a section which has no zeroes over the n-skeleton of M and which is transverse to the zero section of E. Then consider the map (where the norm is take with respect to a euclidean bundle metric on E). The homotopy class of g in π n (S n−1 ) is the obstruction to extend a no where vanishing section over the n-skeleton to the (n + 1)skeleton of M . Since π n (S n−1 ) is isomorphic to the stable homotopy group π S 1 we consider the homotopy class of g as an element therein.
From Lemma 5.4 we infer that the [g] ∈ π S 1 ∼ = Ω fr 1 is equal to the framed divisor κ(E) of E defined by σ, thus E admits a no where vanishing section in case e(E) = 0 and κ(E) = 0. Example 5.6. As an application of our theory we will reprove the following fact due to Whitehead [20] and Eckmann [5]: The number of linear independent vector fields on S 4k+1 is equal to 1 (see also [1] and in [19]).
Denote by ·, · the standard euclidean product in R 4k+2 . The vector field v : R 4k+2 → R 4k+2 , v(x 1 , x 2 , . . . , x 4k+2 ) = (−x 2 , x 1 , . . . , −x 4k+2 , x 4k+1 ) defines a nowhere vanishing vector field on S 4k+1 since v(x), x = 0 for x ∈ S 4k+1 . Let E the subbundle of T S 4k+1 orthogonal to the line bundle spanned by v. For any vector field on S 4k+1 which is in every point linear independent to v there is a nowhere vanishing section of E 2 . Since the Euler class of E vanishes, it suffices to show that κ(E) is not zero by Theorem 5.5 (note that the spin structures of S 4k+1 and that of E are unique up to homotopy). Consider now the vector field Since w(x), x = w(x), v(x) = 0 we have that w is a section of E. Furthermore w is transverse to the zero section of E and the zero locus is given by S = {(x 1 , x 2 , 0, . . . , 0) ∈ S 4k+1 : x 2 1 + x 2 2 = 1}.
In Example 3.10 we saw that T S 4k+1 | S inherits the standard framing from the spin structure. But the induced framing on E| S cannot be the standard framing. To see this assume it inherits the standard framing and let τ 1 , . . . , τ n be a trivialization of E| S , then, since the spin structure on E is induced by T S 4k+1 and v, the map S → SO(4k + 2), x → (x, v(x), τ 1 (x), . . . , τ n (x)) has to be nullhomotopic cf. Example 3.10 (note that v| S is tangent to S) which is a contradiction. Thus from Example 3.10 we deduce that the index of the framed divisor is not zero, hence κ(E) = 1 and therefore E does not admit a nowhere vanishing section from Theorem 5.5.
Remark 5.7. In [6, Theorem 1.6] the authors show, that for any n-dimensional CW-complex of dimension X and any k-dimensional integral cohomology class a ∈ H k (X; Z) there exists an oriented vector bundle over X whose Euler class equals 2 · N (n, k) · a.
Suppose dim X = 2k + 1. By Steenrod's exact sequence (ST) it follows that the Hurewicz map π n (X) → H n (X; Z) is surjective. Then for every a ∈ H n (X; Z) there is a map f a ∈ π n (X) such that f * a (σ) = a, where σ ∈ H n (S; Z) denotes the generator such that 2σ equals to the Euler class of the tangent bundle T S n of S n . Clearly the vector bundle f * a (T S n ) has Euler class 2 · a and therefore N (2k, 2k + 1) = 1 in the notation of [6].
Note that any vector bundle over S n for n = 2, 4, 8 has an Euler class divisible by 2, cf. [2,11]. In the cases n = 2, 4, 8 there are real vector bundles whose Euler class is a generator of H n (S n ; Z), namely the associated bundles to the Hopf fibrations S 2n−1 → S n . We deduce Proposition 5.8. Suppose n = 4 or n = 8 and let M be a (n + 1)-dimensional closed spin manifold. Denote by Vect n (M ) the set oriented vector bundles over M of rank n up to isomorphism. Let E 0 → S n denote the oriented rank n vector bundle such that the Euler class of E 0 is a generator of H n (S n ; Z). Then the map π n (M ) → Vect n (M ), f → f * (E 0 ) is injective.
Proof. We consider f 1 , f 2 ∈ π n (M ) such that E 1 := f * 1 (E 0 ) ∼ = f * 2 (E 0 ) =: E 2 since they represent the Euler class the respective bundles. This implies f * 1 (σ) = f * 2 (σ) for a generator in H n (S n ; Z). Thus it remains to show that κ(f 1 ) = κ(f 2 ). Let x i ∈ S n be a regular value for f i for i = 1, 2. There is a section σ 0,i : S n → E 0 which is transverse to the zero section with an isolated zero in x i (note that the Poincaré dual of x i in S n represents the Euler class of E 0 . Therefore σ 0,i can only exist since if the Euler class is a generator, since the index of transverse sections is always ±1). Then σ i := f * (σ 0,i ) is a transverse section of E i . Note that from the Pontryagin-Thom construction we may assume that f −1 i (x i ) is connected, hence the zero locus of σ i coincides with f −1 i (x i ). Moreover the framed divisor of E i coincides with the degree of f i (cf. Definitions 3.8 and 5.2). Since E 1 ∼ = E 2 we have κ(E 1 ) ∼ = κ(E 2 ) by construction of the framed divisor and Proposition 5.1. From f * 1 (σ) = f * 2 (σ) and κ(f 1 ) = κ(E 1 ) = κ(E 2 ) = κ(f 2 ) it follows from Theorem 4.1 that f 1 is homotopic to f 2 .