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

A Correction to this article was published on 14 September 2021

This article has been updated

Abstract

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. Using homotopy theory it is known that \(\pi ^n(M) = H^n(M;{\mathbb {Z}}) \oplus {\mathbb {Z}}_2\). 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 similar result to the mod 2 degree theorem for maps \(M \rightarrow S^{n+1}\). Finally we will observe that the zero locus of a section in an oriented rank n vector bundle \(E \rightarrow M\) 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 [16] the (stable) homotopy group \(\pi _{n+1}(S^n)\) (\(n\ge 3\)) using differential topology. Let us describe briefly his construction, since this paper will generalize his idea.

He showed that \(\pi _{n+1}(S^n)\) is isomorphic to the bordism group of closed 1-dimensional submanifolds of \({\mathbb {R}}^{n+1}\) furnished with a framing on its normal bundle (a framing is a homotopy class of trivializations, see Sect. 2). We denote this bordism group by \(\Omega _1^{\text {fr}}({\mathbb {R}}^{n+1})\). Let \((C,\varphi )\) be a representative of an element of \(\Omega _1^{\text {fr}}({\mathbb {R}}^{n+1})\), i.e. C is a union of embedded circles in \({\mathbb {R}}^{n+1}\) and there are maps \(\varphi _1,\ldots ,\varphi _n :C \rightarrow {\mathbb {R}}^{n+1}\) such that \((\varphi _1(x),\ldots ,\varphi _n(x))\) is a basis of \(\nu (C)_x\) for every \(x \in C\). Let \(\varphi _{n+1}\) be a trivialization of the tangent bundle of C such that \((\varphi _1(x),\ldots ,\varphi _{n+1}(x))\) is a positive oriented basis of \({\mathbb {R}}^{n+1}\) for every \(x \in C\). Without loss of generality we may assume that \(\varphi _1,\ldots ,\varphi _{n+1}\) is pointwise an orthonormal basis. If \((e_1,\ldots ,e_{n+1})\) denotes the standard basis of \({\mathbb {R}}^{n+1}\), then consider the map \(A=(a_{ij}) :C \rightarrow \mathrm {SO}(n+1)\) such that

$$\begin{aligned} \varphi _i(x) = \sum _{j=1}^{n+1} a_{ij}(x)e_j \end{aligned}$$

for \(x \in C\). Let \(\pi _1(\mathrm {SO}(n+1))\) be identified with \({\mathbb {Z}}_2\), then Pontryagin defines [16, Theorem 20]

$$\begin{aligned} \delta (C,\varphi ) := [A] + (n(C) \mod 2) \end{aligned}$$

where [A] denotes the homotopy class of A in \(\pi _1(\mathrm {SO}(n+1))\) and n(C) is the number of connected components of C. He showed that \(\delta \) is well-defined on \(\Omega _1^{\text {fr}}({\mathbb {R}}^{n+1})\) 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 \(\pi ^n(X):=[X,S^n]\), cf. [3, 18]. The set \(\pi ^n(X)\) for X a finite CW complex of dimension \(n+1\) carries naturally a group structure, which is described in the beginning of Sect. 4. Steenrod showed [19, Theorem 28.1, p. 318] that \(\pi ^n(X)\) fits into a short exact sequence

$$\begin{aligned} 0 \longrightarrow H^{n+1}(X;{\mathbb {Z}}_2)/ \text {Sq}^2 \mu (H^{n-1}(X;{\mathbb {Z}})) \longrightarrow \pi ^n(X) \longrightarrow H^n(X;{\mathbb {Z}}) \longrightarrow 0, \end{aligned}$$

where \(\mu :H^{*}(X;{\mathbb {Z}}) \rightarrow H^*(X;{\mathbb {Z}}_2)\) is the mod 2 reduction homomorphism (see also [17] for a more geometric proof of this sequence). Here the surjective map is the Hurewicz homomorphism which assigns to every \(f \in \pi ^n(X)\) the cohomology class \(f^*(\sigma ) \in H^n(X;{\mathbb {Z}})\) where \(\sigma \in H^n(S^n,{\mathbb {Z}})\) is a fixed generator.

Moreover using methods of Larmore and Thomas [11] Taylor showed in [20, Theorem 6.2, Example 6.3] that the short exact sequence splits, provided the images of \(\text {Sq}^2 :H^{n-1}(X;{\mathbb {Z}}_2) \rightarrow H^{n+1}(X;{\mathbb {Z}}_2)\) and \(\text {Sq}^2 \circ \mu :H^{n-1}(X;{\mathbb {Z}}) \rightarrow H^{n+1}(X;{\mathbb {Z}}_2)\) coincide.

If \(X=M\) is an oriented manifold then the second Wu class [24] is equal to the second Stiefel-Whitney class \(w_2(M)\), hence \(\text {Sq}^2(x) = w_2(M)\smile x\) for \(x \in H^{n-1}(M;{\mathbb {Z}}_2)\). Therefore if M is spin then \(\pi ^n(M)\) fits into the exact sequence

$$\begin{aligned} 0 \longrightarrow {\mathbb {Z}}_2 \longrightarrow \pi ^n(M) \longrightarrow H^n(M;{\mathbb {Z}}) \longrightarrow 0. \end{aligned}$$
(ST)

and (ST) splits by [20, Example 6.3] thus

$$\begin{aligned} \pi ^n(M) \cong H^n(M;{\mathbb {Z}}) \oplus {\mathbb {Z}}_2 \end{aligned}$$

as abelian groups. However the splitting map is constructed in a purely homotopy theoretic way and an aim of this article is to provide a geometric description in case M is a spin manifold.

This splitting map \(\kappa :\pi ^n(M) \rightarrow {\mathbb {Z}}_2\) (see Definition 3.8) for (ST) will be constructed similarly to Pontryagin’s invariant \(\delta \) from above. An important ingredient in Pontryagin’s construction was the canonical background framing given by the standard basis of \({\mathbb {R}}^{n+1}\), which allowed him to define the map \(A:C \rightarrow \mathrm {SO}(n+1)\). In general if we replace \(S^{n+1}\) or \({\mathbb {R}}^{n+1}\) 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 \(\pi ^n(M) \rightarrow H^n(M;{\mathbb {Z}})\). Finally we show that the splitting map possesses a naturality property, cf. Proposition 4.3 and that for a map \(f:M\rightarrow S^n\) the number \(\kappa (f)\) can be described by a counting formula, cf. Corollary 4.4. This is an analogous result to the mod 2 Hopf theorem, see [15, §4]. It should be mentioned that in [8] the authors discuss the case \(n=3\) and in [10] a similar construction of a \({\mathbb {Z}}_2\) invariant was used to classify quaternionic line bundles over closed spin 5-manifolds.

In Sect. 5 we will apply the results of Sects. 3 and 4 to the theory of vector bundles. Suppose \(E \rightarrow M\) is an 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 of its zero locus an element of \(\Omega _1^{\text {fr}}(M)\) and this element is independent of the transverse section. Thus using \(\kappa \) one defines an invariant \(\kappa (E) \in \Omega _1^\text {fr}\) of the isomorphism class of the bundle \(E\rightarrow M\). In Theorem 5.5 it is shown that \(\kappa (E)\) 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 \(\pi ^n(M)\) 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.

Preliminaries

If not otherwise stated we denote by M an \((n+1)\)-dimensional oriented, closed and connected manifold, where \(n\ge 3\). Let N be a arbitrary manifold and \(E \rightarrow N\) a trivial vector bundle over N of rank r. A trivialization of \(E\rightarrow N\) are r sections \(s_1,\ldots ,s_r :N \rightarrow E\), such that \((s_1(q),\ldots ,s_r(q))\) is a basis of the fiber \(E_q\) for all \(q \in N\). A framing \(\varphi \) of \(E\rightarrow N\) is a homotopy class of trivializations.

We recall now the notion of bordism classes of normally framed submanifold in M of dimension k (cf. [15, §7]). Let C be a k-dimensional closed submanifolds of M. We say that C is normally framed if the normal bundle of C is trivial and possesses a framing \(\varphi \). Two such normally framed submanifolds \((C_0,\varphi _0)\) and \((C_1,\varphi _1)\) are framed bordant if there is a \((k+1)\)-dimensional submanifold \(\Sigma \subset M \times [0,1]\) such that

  1. (a)

    \(\partial \Sigma \cap (M \times i) = C_i\) for \(i=0,1\),

  2. (b)

    \(\partial \Sigma = C_0 \cup C_1\),

  3. (c)

    \(\Sigma \) is normally framed in \(M \times [0,1]\) such that the framing restricted to the \(\partial \Sigma \cap (M \times i)\) coincides with \(\varphi _i\).

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 in M denoted by \(\Omega _k^{\text {fr}}(M)\). If \((C,\varphi )\) is a normally framed submanifold then we denote by \([C,\varphi ]\) its bordism class in \(\Omega _k^{\text {fr}}(M)\).

The Pontryagin-Thom map provides a bijection between \(\pi ^{n+1-k}(M)\) and \(\Omega _k^{\text {fr}}(M)\) as follows (cf. [15, §7]): Fix an orientation on \(S^n\) and let \(f :M \rightarrow S^{n+1-k}\) represent an element of \(\pi ^{n+1-k}(M)\). Choose a regular value \(x_0 \in S^{n+1-k}\) and set \(C_{x_0}:= f^{-1}(x_0)\). Moreover choosing an oriented basis of the tangent space \(T_{x_0}S^{n+1-k}\) endows the normal bundle with a framing \(\varphi _{x_0}\) by means of the derivative of f. The bordism class \([C_{x_0},\varphi _{x_0}] \in \Omega _{k}^{\text {fr}}(M)\) is well defined and the map

$$\begin{aligned} \pi ^{n+1-k}(M)\longrightarrow \Omega _k^{\text {fr}}(M), \quad [f]\mapsto [C_{x_0},\varphi _{x_0}]. \end{aligned}$$

is a bijection, see [15, Theorem B and A].

A stable framing of a real vector bundle \(E \rightarrow C\) of rank r is an equivalence class of trivializations of

$$\begin{aligned} E \oplus \varepsilon ^l \end{aligned}$$

for some \(l \in {\mathbb {N}}\) where two trivializations

$$\begin{aligned} \tau _1 :E \oplus \varepsilon ^{l_1} \rightarrow \varepsilon ^{r+l_1} \quad \text {and}\quad \tau _2 :E \oplus \varepsilon ^{l_2} \rightarrow \varepsilon ^{r+l_2}, \end{aligned}$$

are considered to be equivalent if there exists some \(L>l_1,l_2\) such that the isomorphisms

$$\begin{aligned} \tau _1 \oplus \text {id}:E \oplus \varepsilon ^{l_1} \oplus \varepsilon ^{L-l_1} \rightarrow \varepsilon ^{L+r} \end{aligned}$$

and

$$\begin{aligned} \tau _2 \oplus \text {id}:E \oplus \varepsilon ^{l_2} \oplus \varepsilon ^{L-l_2} \rightarrow \varepsilon ^{L+r} \end{aligned}$$

are homotopic, cf. [5, Section 8.3]. If E is the tangent bundle of C, then a stable framing of TC is called a stable tangential framing. If E is the normal bundle of an embedding of C into a sphere whose dimension is big enough, then we call a stable framing a stable normal framing.

We define \(\Omega _k^{\text {fr}}\) to be the bordism classes of stably (tangential) framed k-dimensional manifolds. More precisely two stably framed manifolds \((C_0,\varphi _0)\) and \((C_1,\varphi _1)\) where \(\varphi _i :TC_i \oplus \varepsilon ^l \rightarrow \varepsilon ^{k+l}\) is an isomorphism are equivalent if there is a bordism \(\Sigma \) between \(C_0\) and \(C_1\) such that the tangent bundle of \(\Sigma \) possesses a stable framing and the restriction on \(C_0\) and \(C_1\) coincides with the framing \(\varphi _0\) and \(\varphi _1\) respectively. Note that \(\Omega _k^{\text {fr}}\) is isomorphic to \(\pi _k^S\), the k-stable homotopy group of spheres (cf. [5, Theorem 8.17]) and by the Pontryagin-Thom construction we have \(\Omega _k^{\text {fr}} = \varinjlim _l \Omega _k^{\text {fr}}(S^l)\) where we use the equatorial embeddings \(S^{l_1} \hookrightarrow S^{l_2}\) if \(l_1 < l_2\) to construct well-defined maps \(\Omega _k^{\text {fr}}(S^{l_1}) \rightarrow \Omega _k^{\text {fr}}(S^{l_2})\).

For this article the case \(k=1\) will be of importance. In this case we have \(\Omega _1^{\text {fr}}\cong \pi _1^S \cong {\mathbb {Z}}_2\). Consider a connected and closed 1-dimensional manifold \(S_0\) and stable tangential framing

$$\begin{aligned} \varphi _0 :TS_0 \oplus \varepsilon ^n {\mathop {\longrightarrow }\limits ^{\sim }} \varepsilon ^{n+1}. \end{aligned}$$

From the discussion above, \((S_0,\varphi _0)\) defines a class in \(\Omega _1^{\text {fr}}\) and can be realized as follows: Let \(S_0 =\{(x_1,\ldots ,x_{n+1}) \in {\mathbb {R}}^{n+1} : x_1^2+x_2^2=1,\, x_i=0,\, i=3,\ldots ,n+1\}\). Denote by \(e_1,\ldots ,e_{n+1}\) the canonical basis of \({\mathbb {R}}^{n+1}\) and \(E_i(x)=e_i\) for \(x\in {\mathbb {R}}^{n+1}\). Moreover let \(V(x)=x\) for \(x \in {\mathbb {R}}^n\). The normal bundle \(\nu (S_0)\) of \(S_0\) is trivialized by \(V,E_3,\ldots ,E_{n+1}\) restricted to \(S_0\). Using this normal framing we obtain a stable framing

$$\begin{aligned} TS_0 \oplus \varepsilon ^{n} \cong TS_0 \oplus \nu (S_0) \cong (T{\mathbb {R}}^{n+1})|_{S_0} \cong \varepsilon ^{n+1} \end{aligned}$$

where the latter framing is induced by \(E_1,\ldots ,E_{n+1}\). Hence this defines an element in \(\Omega _1^{\text {fr}}(S^{n+1})\) which represents the framed null bordism, since the framing of \(\nu (S_0)\) can be extended to a properly embedded stably framed disc in \(S^{n+1} \times [0,1]\). Clearly the non-trivial element of \(\Omega _1^{\text {fr}}(S^{n+1})\) can be represented by twisting the normal framing \(V, E_3,\ldots ,E_{n+1}\) with a map \(S_0 \rightarrow \text {SO}(n)\) such that its homotopy class in \(\pi _1\left( \text {SO}(n) \right) \cong {\mathbb {Z}}_2\) is not zero. Every stable tangential framing of a closed and connected 1-dimensional manifold can be obtained in this way.

Let \(E \rightarrow N\) be an oriented vector bundle over a manifold N. After choosing a Euclidean bundle metric on E one obtains a vector bundle with structure group \(\mathbf{SO }(n)\). Since the space of such Euclidean bundle metrics is contractible, every construction which depends up to homotopy from a metric is independent thereof.

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 \(N \rightarrow B \text {SO}(n)\) to a map \(N \rightarrow B \text {Spin}(n)\) in the fibration \(K({\mathbb {Z}}_2,1) \rightarrow B \text {Spin}(n) \rightarrow B \text {SO}(n)\). Consequently E is a spin bundle if it is spinnable and a spin structure is fixed. If a spin structure is fixed on \(E \rightarrow N\) then any other spin structure is in 1 : 1 correspondence with elements in \(H^1(N;{\mathbb {Z}}_2)\), cf. [12, Chapter II, Theorem 1.7]

We write F(N) for the orthonormal frame bundle of an oriented manifold N, where the frames are consistent with the given orientation on N. If \(V\subset N\) is a submanifold such that its normal bundle is framed then we obtain an embedding \(F(V) \subset F(N)\). Thus a spin structure on N induces a spin structure on V, cf. [14]. In particular if V is the boundary of a spin manifold N, then V inherits a spin structure from N. Finally if \(E \rightarrow N\) is a vector bundle with a spin structure and \(V\subset N\) a submanifold, then clearly \(E|_V \rightarrow V\) also inherits a spin structure from \(E \rightarrow N\).

Let \(E \rightarrow S^1\) be a spinnable vector bundle of rank \(r \ge 3\) over the unit circle \(S^1\). Then E has exactly two non-isomorphic spin structures. Clearly \(E\rightarrow S^1\) can be extended to \(E \rightarrow D^2\), where \(D^2\) denotes the closed unit disc in \({\mathbb {R}}^2\) . Since \(D^2\) is contractible \(E\rightarrow D^2\) admits a unique spin structure. Restricting this structure to the boundary of \(D^2\) gives a spin structure on \(E\rightarrow S^1\), which will be called the standard spin structure (\(D^2\) is equipped with the standard orientation of \({\mathbb {R}}^3\) and the orientation on \(S^1\) is induced by the outward pointing normal of \(S^1\)). The other should be called the non-standard spin structure. In other words, the standard spin structure on \(E \rightarrow S^1\) can be extended to \(D^2\) in contrary to the non-standard one.

The index of framed circles

In this section we define the key invariant of this article. For its construction the following basic lemma is the crucial observation.

Lemma 3.1

Let \(E \rightarrow S^1\) be a spinnable vector bundle of rank \(\ge 3\). 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 \(\mathbf {Spin}(n)\)-principal bundle over \(S^1\) which is a two-sheeted cover over the frame bundle F(E) of E. Let \(\pi :F'(E)\rightarrow F(E)\) be the projection which is equivariant with respect to the two-sheeted covering \(\mathbf {Spin}(n)\rightarrow \mathbf{SO }(n)\). Clearly \(F'(E)\) is the trivial \(\mathbf {Spin}(n)\)-principal bundle over \(S^1\) and denote by \(\sigma :S^1 \rightarrow F'(E)\) a global section. Then \(\pi \circ \sigma \) is a global section of F(E) hence a trivialization of \(E \rightarrow S^1\). Any other such global section \({{\widetilde{\sigma }}} :S^1 \rightarrow F(E)\) differs from \(\sigma \) by a map \(\varphi :S^1 \rightarrow \mathbf {Spin}(n)\). Since \(\pi _1(\mathbf {Spin}(n))=1\) the map \(\varphi \) has to be null-homotopic which means that the two trivializations \(\pi \circ \sigma \) and \(\pi \circ {{\widetilde{\sigma }}}\) have to be homotopic, thus they define the same framing on E. \(\square \)

In the same way one proves

Corollary 3.2

Let \(\Sigma \) be a 1-dimensional CW-complex (not necessarily connected) and \(E\rightarrow \Sigma \) a vector bundle of rank \(\ge 3\) 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 \(E \rightarrow S^1\) be a spinnable vector bundle. The framing induced by the standard spin structure on E is called the standard framing and the one induced by the 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 \({\mathbb {R}}^{n+2}\) which admits a unique spin structure.

Let \(S_0 \subset S^{n+1}\) be the intersection of a 2-dimensional linear subspace \(W\subset {\mathbb {R}}^{n+2}\) with \(S^{n+1}\) and denote by \(D_0^2 = W \cap D^{n+2}\). Thus after Lemma 3.1\(TS^{n+1}|_{S_0}\) inherits a framing from the spin structure. Denoting by \(\varphi _1,\ldots ,\varphi _{n+1}\) a trivialization of this framing, the framing

$$\begin{aligned} {\overline{\varphi }}:S_0 \rightarrow \text {SO}(n+2),\quad x \mapsto (x,\varphi _1(x),\ldots ,\varphi _{n+1}(x)) \end{aligned}$$

must be null homotopic in \(\text {SO}(n+2)\) by the definition of the spin structures of \(S^{n+1}\) and \(TS^{n+1}|_{S_0}\) (such that it lifts to \(\text {Spin}(n+2)\)). Thus \({\overline{\varphi }}\) must be homotopic the constant framing \(x \mapsto (e_1,\ldots ,e_{n+2})\), where \(e_1,\ldots ,e_{n+2}\) denotes the canonical basis of \({\mathbb {R}}^{n+2}\). In particular this means that \(TS^{n+1}|_{S_0}\) inherits the standard framing from the spin structure of \(S^{n+1}\).

\(\Omega _1^{\text {fr}}(M)\) 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, they are framed bordant in M to framed submanifolds \({\tilde{C}}\) and \({\tilde{C}}'\) with empty intersection. Taking the equivalence class of the disjoint union \({\tilde{C}} \cup {\tilde{C}}'\) with the respective framings yields an abelian group structure on \(\Omega _1^{\text {fr}}(M)\), cf. [15, Problem 17 and p. 50].

Next, we construct a homomorphism \(\kappa :\Omega _1^{\text {fr}}(M) \rightarrow \Omega _1^{\text {fr}}\) for M a spin manifold. Therefore let \((C,\varphi _C)\) be a closed submanifold of dimension 1, such that its normal bundle \(\nu (C)\) is framed by \(\varphi _C\) (thus representing an element in \(\Omega _1^{\text {fr}}(M)\)). From Lemma 3.1 the bundle \(TM|_C\) inherits a framing \(\varphi _\sigma \) from the spin structure of M. Using also the framing of \(\varphi _C\) we obtain a stable tangential framing

$$\begin{aligned} \varepsilon ^{n+1} \cong TM|_C \cong TC \oplus \nu (C) \cong TC \oplus \varepsilon ^n \end{aligned}$$

which we denote by \(\varphi _{\text {st}}\).

Proposition 3.5

The bordism class \([C,\varphi _{\text {st}}] \in \Omega _1^{\text {fr}}\) depends only on the bordism class \([C,\varphi _C] \in \Omega _1^{\text {fr}}(M)\).

Proof

Let \((C',\varphi _{C'})\) be another normally framed closed 1-dimensional submanifold framed bordant to \((C,\varphi _C)\). Thus there is a bordism \(\Sigma \subset M \times I\) between C and \(C'\) such that the normal bundle of \(\Sigma \) in \(M\times I\) possess a framing \(\varphi _\Sigma \). By definition restricting \(\varphi _\Sigma \) to C and \(C'\) yields \(\varphi _C\) and \(\varphi _{C'}\) respectively. Since \(\Sigma \) is homotopy equivalent to a 1-dimensional CW-complex and since \(M\times I\) inherits a unique spin structure from M we obtain a framing \(\varphi _{\Sigma ,\sigma }\) on \(T(M\times I)|_{\Sigma }\). Of course the framings \(\varphi _{\Sigma ,\sigma }\) restricted to C and \(C'\) are just the framings \(\varphi _\sigma \) and \(\varphi _\sigma '\) respectively (i.e. induced by the spin structure of \(TM|_C\) and \(TM|_{C'}\)). Since

$$\begin{aligned} T(M\times I)|_\Sigma \cong T\Sigma \oplus \nu (\Sigma ) \end{aligned}$$

the framings \(\varphi _{\Sigma ,\sigma }\) and \(\varphi _\Sigma \) determine a stable framing \(\varphi _{\Sigma ,\text {st}}\) of \(T\Sigma \). Then \((\Sigma ,\varphi _{\Sigma ,\text {st}})\) is a stably framed bordism between \((C,\varphi _{\text {st}})\) and \((C',\varphi _{\text {st}}')\). \(\square \)

Remark 3.6

As described above, the group structure of \(\Omega _{1}^{\text {fr}}(M)\) is given by disjoint union of submanifolds and their respective normal framings. Let \((C,\varphi )\) be a framed 1-dimensional closed submanifold of M and denote by \(C=S_1 \cup \ldots \cup S_k\) the connected components of C. We may assume that the union is always disjoint. Thus \(S_i\) is an embedded circle and \(\varphi _i := \varphi |_{S_{i}}\) a normal framing of \(S_i\). Consequently we have

$$\begin{aligned}{}[C,\varphi ] = \sum _{i=1}^{k}\; [S_i,\varphi _i] \end{aligned}$$

in \(\Omega _1^{\text {fr}}(M)\).

Definition 3.7

Let \(S \subset M\) be an embedded circle and \(\varphi \) a framing of \(\nu (S)\). We call the bordism class \([S,\varphi ] \in \Omega _1^{\text {fr}}(M)\) a framed circle of M. The corresponding stable class \([S,\varphi _{\text {st}}]\in \Omega _1^{\text {fr}}\) will be called the index of \([S,\varphi ]\) (with respect to the spin structure of M) and will be denoted by \({\text {ind}}(S,\varphi )\).

Definition 3.8

Let M be an \((n+1)\)-dimensional closed spin manifold. Then we define a map

$$\begin{aligned} \kappa :\Omega _1^{\text {fr}}(M) \rightarrow \Omega _1^{\text {fr}},\quad [C,\varphi ] \mapsto \kappa \left( [C,\varphi ] \right) := \sum _{\begin{array}{c} S \subset C,\\ S\text { connected} \end{array}}^{} {\text {ind}}(S,\varphi |_S) = [C,\varphi _{\text {st}}]. \end{aligned}$$

We call \(\kappa \) the degree map of M with respect to the chosen spin structure.

Remark 3.9

It follows from the construction that \(\kappa \) is a homomorphism.

Examples 3.10

  1. (a)

    Consider \(S^{n+1}\) with the induced spin structure from the unit disc of \({\mathbb {R}}^{n+2}\), cf. Example 3.4 respectively [14]. Let \(S_0\) be the intersection of \(S^{n+1}\) with a 2-dimensional linear subspace W of \({\mathbb {R}}^{n+2}\). We argued in Example 3.4 that \(TS^{n+1}|_{S_0}\) inherits the standard framing. Choose the standard framing \(\varphi _0\) on \(\nu (S_0)\), then

    $$\begin{aligned} \kappa ([S_0,\varphi _0])=0. \end{aligned}$$

    Consequently the non-standard framing \(\varphi _1\) of \(\nu (S_0)\) yields

    $$\begin{aligned} \kappa ([S_0,\varphi _1])\ne 0. \end{aligned}$$
  2. (b)

    Let N be a closed, simply connected, spin manifold of dimension n. Then \(M:=S^1 \times N\) admits two different spin structures since \(H^1(S^1 \times N;{\mathbb {Z}}_2) \cong H^1(S^1;{\mathbb {Z}}_2) \cong {\mathbb {Z}}_2\). M is the boundary of \(D^2 \times N\) 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 \(D^2 \times N\) and the other not. We call the former one the standard spin structure and the latter one the non-standard spin structure of \(S^1 \times N\).

    For \(q_0 \in N\) consider the circle \(S_0:= S^1 \times q_0 \subset S^1 \times N\). Clearly we have a canonical isomorphism

    $$\begin{aligned} \nu (S_0) \cong S_0 \times T_{q_0}N. \end{aligned}$$

    Thus choosing a basis in \(T_{q_0}N\) gives a framing \(\varphi _0\) on \(\nu (S_0)\) which extends to a framing of \((D^2 \times q_0) \times T_{q_0}N\). This implies

    $$\begin{aligned} \kappa _0([S_0,\varphi _0]) =0 \end{aligned}$$

    for the standard spin structure and

    $$\begin{aligned} \kappa _1([S_0,\varphi _0]) \ne 0 \end{aligned}$$

    for the non-standard spin structure.

    For \(q_1 \in N\) with \(q_0 \ne q_1\) we consider \(C = S^1 \times q_0 \cup S^1 \times q_1\) with fixed normal framing on \(S^1 \times q_i\) which gives a framing \(\varphi \) on C. Then \(\kappa ([C,\varphi ])\) is independent of the chosen spin structure of M. This shows that in general \(\kappa \) will depend on the spin structure. The next proposition will show how it depends on it.

Suppose \(C\subset M\) is a closed 1-dimensional submanifold. Then C defines a \({\mathbb {Z}}_2\) fundamental homology class \([C] \in H_1(M;{\mathbb {Z}}_2)\). We denote by \(w(C) \in H^n(M;{\mathbb {Z}}_2)\) the cohomology class which is the Poincaré dual of [C].

Proposition 3.11

Fix a spin structure \(\sigma \) on M and denote by \(\kappa \) the degree map of M with respect to the chosen spin structure. Choose another spin structure of M, which is represented by \(\alpha \in H^1(M;{\mathbb {Z}}_2)\) and denote by \(\kappa ^\alpha \) the corresponding degree map. Then we have

$$\begin{aligned} \kappa ([C,\varphi ]) = \kappa ^\alpha ([C,\varphi ]) + \delta (\alpha \smile w(C)), \end{aligned}$$

where \(\delta :H^{n+1}(M;{\mathbb {Z}}_2) \rightarrow \Omega _1^{\text {fr}}\) is the unique isomorphism. Thus if \(w(C) \smile \alpha =0\) then \(\kappa \left( [C,\varphi ] \right) =\kappa ^{\alpha }\left( [C,\varphi ] \right) \).

Proof

Assume first that \((S,\varphi )\) is a framed circle and \(i :S \rightarrow M\) is the inclusion. The spin structure \(\sigma \) induces a spin structure on \(TM|_S = i^*(TM)\) and the spin structure induced by \(\alpha \) is represented by \(i^*(\alpha ) \in H^1(S;{\mathbb {Z}}_2)\). Of course \(TM|_S\) can have at most two different spin structures. From the definition of the index we have

$$\begin{aligned} {\text {ind}}(S,\varphi ) = {\text {ind}}_\alpha (S,\varphi ) + \delta (i^*(\alpha )) \end{aligned}$$

where \({\text {ind}}\) is defined by \(\sigma \), \({\text {ind}}_\alpha \) by \(\alpha \) and \(\delta :H^1(S;{\mathbb {Z}}_2) \rightarrow \Omega _1^{\text {fr}}\) the unique isomorphism.

Let \([S] \in H_1(S;{\mathbb {Z}}_2)\) be the \({\mathbb {Z}}_2\) fundamental class of S, then \(i^*(\alpha )\frown [S] \in H_0(S;{\mathbb {Z}}_2)\), which is mapped under \(i_*\) to \(\alpha \frown i_*([S]) \in H_0(M;{\mathbb {Z}}_2)\). Let \([M] \in H_n(M;{\mathbb {Z}}_2)\) denote the \({\mathbb {Z}}_2\) fundamental class of M. Then

$$\begin{aligned} \alpha \frown i_*([S]) = \alpha \frown \left( w(S) \frown [M] \right) =\left( \alpha \smile w(S) \right) \frown [M], \end{aligned}$$

where we used that \(i_*([S])\) is Poincaré dual to w(S). Since \( \cdot \frown [S]\) and \( \cdot \frown [M]\) are isomorphisms by Poincaré duality as well as \(i_*:H_0(S;{\mathbb {Z}}_2) \rightarrow H_0(M;{\mathbb {Z}}_2)\) because S and M are connected we infer

$$\begin{aligned} {\text {ind}}(S,\varphi ) = {\text {ind}}_\alpha (S,\varphi ) + \delta (\alpha \smile w(S)) \end{aligned}$$

where now \(\delta :H^{n+1}(M;{\mathbb {Z}}_2) \rightarrow \Omega _1^{\text {fr}}\) is again the unique isomorphism.

Consider now \((C,\varphi )\) with the disjoint union \(C= S_1 \cup \ldots S_k\) and \(\varphi _j := \varphi |_{S_j}\), such that \(S_j\) is connected. With the previous computations we have

$$\begin{aligned} \kappa ([C,\varphi ]) =\sum _{j=1}^{k} \left( {\text {ind}}_\alpha (S_j,\varphi _j) +\delta (\alpha \smile w(S_j) \right) ) = \kappa ^\alpha ([C,\varphi ]) + \delta (\alpha \smile w(C)). \end{aligned}$$

and the proposition follows. \(\square \)

We continue with the description of the ,,dual” short exact sequence to (ST). There is a natural group homomorphism \(\Omega _1^{\text {fr}}(M) \rightarrow \Omega _1^{\text {SO}}(M)\), which assigns to every framed 1-submanifold \([C,\varphi ]\) the oriented bordism class induced by the orientation of framing \(\varphi \). 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 [21] we have an isomorphism

$$\begin{aligned} \Omega _1^{\text {SO}}(M) \rightarrow H_1(M;{\mathbb {Z}}) \end{aligned}$$

which assigns every oriented submanifold its fundamental class in \(H_1(M;{\mathbb {Z}})\). Thus we obtain a group homomorphism

$$\begin{aligned} \Phi :\Omega _1^{\text {fr}}(M) \rightarrow H_1(M;{\mathbb {Z}}) \end{aligned}$$
(1)

which is clearly surjective. The kernel of \(\Phi \) is at most isomorphic to \({\mathbb {Z}}_2\) and elements of the kernel are represented by framed circles \((S,\varphi )\) such that S is oriented null-bordant, i.e. there is an embedded oriented disc \(D\subset M \times I\) with the properties \(\partial D =S\) and the orientations of \(\partial D\) 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 \({\mathbb {Z}}_2\).

Lemma 3.12

The restricted degree map \(\kappa |_{\ker \Phi } :\ker \Phi \rightarrow \Omega _1^{\text {fr}}\) is an isomorphism.

Proof

Since \(\kappa \) is a homomorphism it will map the neutral element of \(\ker \Phi \) to that of \(\Omega _1^{\text {fr}}\). Thus it suffices to show the following: Let \((S,\varphi )\) be a framed circle such that S is oriented null-bordant in M but \(\varphi \) cannot be extended over the nullbordism. We have to show \(\kappa ([S,\varphi ])\ne 0\), where 0 denotes the neutral element of \(\Omega _1^{\text {fr}}\). We may assume that S lies in a chart of M.Footnote 1 Thus we may embed S into \({\mathbb {R}}^{n+1}\) endowed with a normal framing, which cannot be extended over a nullbordism in \({\mathbb {R}}^{n+1}\). Hence the index of \((S,\varphi )\) defines a non-trivial element in \(\Omega _1^{\text {fr}}\) (note that since \(w(S)=0\) the element \(\kappa [(S,\varphi )]\) does not depend on the spin structure of M, cf. Lemma 3.11). \(\square \)

Thus we may identify \(\ker \Phi \) with \(\Omega _1^{\text {fr}}\) via \((\kappa |_{\ker \Phi })^{-1}\) and we obtain a short exact sequence

$$\begin{aligned} 0 \longrightarrow \Omega _1^{\text {fr}} \longrightarrow \Omega _1^{\text {fr}}(M) \longrightarrow H_1(M;{\mathbb {Z}}) \longrightarrow 0 \end{aligned}$$

and from Lemma 3.12\(\kappa \) is a splitting map. Therefore we showed

Theorem 3.13

Let M be an \((n+1)\)-dimensional closed spin manifold. Choose a spin structure on M. Then

$$\begin{aligned} \Omega _1^{\text {fr}}(M) \longrightarrow H_1(M;{\mathbb {Z}}) \oplus \Omega _1^{\text {fr}}, \quad [C,\varphi ] \mapsto ([C],\kappa ([C,\varphi ])) \end{aligned}$$

is an isomorphism of abelian groups.

We finish this section by giving an alternative way to compute the index of a framed circle in the spirit of Pontryagin [16]. Suppose \([S,\varphi ]\) is a framed circle, thus there are trivializations of \(\nu (S)\) and \(TM|_{S}\) such that we obtain the stable framing

$$\begin{aligned} \varepsilon ^{n+1} \cong TS \oplus \varepsilon ^n \end{aligned}$$

(where we can assume that the isomorphism is orientation preserving). Denote by \(v_1,\ldots ,v_{n+1}\) and by \(w_2,\ldots ,w_{n+1}\) the trivializations of \(TM|_S\) and \(\nu (S)\) respectively. Let \(w_1\) be a trivialization of TS. Let \(\Phi :TS \oplus \varepsilon ^n \rightarrow \varepsilon ^{n+1}\) be the isomorphism of the stable framing, then there is a matrix \(A=(A_{ij}) :S \rightarrow \mathrm {GL}^{+}(n+1)\) (where \(\text {GL}^+(n+1)\) is the set of all invertible real matrices of size \((n+1) \times (n+1)\) with positive determinant) such that

$$\begin{aligned} \Phi (w_i) = \sum _{j=1}^{n+1} A_{ij} \cdot v_j. \end{aligned}$$

Since \(\mathrm {SO}(n+1)\) is a strong deformation retract of \(\mathrm {GL}^+(n+1)\) we have \(\pi _1 \left( \mathrm {GL}^+(n+1)\right) \cong {\mathbb {Z}}_2\). The map \(A :S \rightarrow \mathrm {GL}^+(n+1)\) defines an element \([A] \in \pi _1\left( \mathrm {GL}^+(n+1) \right) \). Changing the homotopy classes of trivializations of \(TM|_S\) and \(\nu (S)\) does not change [A]. Furthermore [A] is also independent of the choice of trivializations of TS.

According to the Preliminaries (Section 2) any stable framing \({\text {ind}}(S,\varphi )\) can be represented by a framed circle \(S_0\) in \({\mathbb {R}}^{n+1}\) such that

$$\begin{aligned} TS_0 \oplus \varepsilon ^n \cong TS_0 \oplus \nu (S_0) \cong (T{\mathbb {R}}^{n+1})|_{S_0} \cong \varepsilon ^{n+1} \end{aligned}$$

recovers the stable framing of \((S,\varphi )\). It follows that

$$\begin{aligned} {\text {ind}}(S,\varphi ) = \delta (S_0,\varphi _0), \end{aligned}$$

where \(\delta \) is the invariant constructed by Pontryagin, [16, Theorem 20]. We will use a different notation: Let us denote by [A] the homotopy class constructed above from the stable framing and by \(\overline{[A]}\) the element \([A]+1 \in \Omega _1^{\text {fr}}(S^n)\cong {\mathbb {Z}}_2\) where 1 is the non-trivial element. Thus we proved

Lemma 3.14

We identify \(\pi _1\left( \mathrm {GL}^+(n+1) \right) \) with \(\Omega _1^{\text {fr}}\) by the unique isomorphism \({\mathbb {Z}}_2 \rightarrow {\mathbb {Z}}_2\). Then

$$\begin{aligned} \overline{[A]} = {\text {ind}}(S,\varphi ). \end{aligned}$$

Computation of \(\pi ^n(M)\)

We start this section to explain the group structure of \(\pi ^n(M)\). Let \(j :S^{n} \vee S^n \rightarrow S^n \times S^n\) be the inclusion of the (\(2n-1\))-skeleton of \(S^n\times S^n\) (endowed with the standard CW structure) then, since M is \(n+1\)-dimensional CW complex, the induced map \(j_\# :[M,S^n\vee S^n] \rightarrow [M,S^n\times S^n]\) is an isomorphism. For \(f,g \in \pi ^n(M)\) the group structure is defined by

$$\begin{aligned} f + g := (\text {id}_{S^n} \vee \text {id}_{S^n})_\# \circ (j_\#)^{-1}(f\times g). \end{aligned}$$

This makes \(\pi ^n(M)\) to an abelian group.

Now, let \(f :M \rightarrow S^n\) be a differentiable map and \(x_0 \in S^n\) a regular value. We orient \(S^n\) by the normal vector field pointing outwards and the standard orientation of \({\mathbb {R}}^{n+1}\).

Let \(\Psi :\pi ^n(M) \rightarrow H^n(M;{\mathbb {Z}})\) be the map \(\Psi ([f]):=f^{*}\sigma \) where \(\sigma \in H^n(S^n;{\mathbb {Z}})\) is a fixed generator. We define the analogous degree map \(\kappa :\pi ^n(M) \rightarrow \pi _1^S\), where \(\pi _1^S\) is the first stable homotopy group of spheres, as follows: \(\kappa \) is the composition of

$$\begin{aligned} \pi ^n(M) \overset{\sim }{\longrightarrow } \Omega _1^{\text {fr}}(M) \overset{\kappa }{\longrightarrow } \Omega _1^{\text {fr}} \overset{\sim }{\longrightarrow } \pi _1^S. \end{aligned}$$

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

  1. (a)

    The generator of \(\ker \Psi \cong {\mathbb {Z}}_2\) is given by the homotopy class of the map \(\eta \circ \omega :M\rightarrow S^{n}\), where \(\eta \) represents a generator of \(\pi _{n+1}(S^n)\) and \(\omega :M \rightarrow S^{n+1}\) is a map of odd degree. Thus \(\ker \Psi \cong \pi _{n+1}(S^n)\).

  2. (b)

    Identifying \(\pi _1^S\) with \(\pi _{n+1}(S^n)\) the degree map \(\kappa :\pi ^n(M) \rightarrow \pi _1^S\) splits the short exact sequence (ST). Thus we have

    $$\begin{aligned} \pi ^n(M) \longrightarrow H^n(M;{\mathbb {Z}}) \oplus \pi _{n+1}(S^n),\quad [f] \mapsto (f^*\sigma ,\kappa ([f])). \end{aligned}$$

    is an isomorphism of abelian groups.

Proof

Clearly we have \([\eta \circ \omega ] \in \ker \Psi \). For (a) it is enough to check that \(\kappa ([\eta \circ \omega ])\) is non-zero in \(\pi _1^S\). We choose an odd degree map \(\omega :M \rightarrow S^{n+1}\) as follows: Let \(\{p_1,\ldots ,p_l\}\) be the preimage of a regular value \(y_0\) and choose open sets \(U_1,\ldots ,U_l \subset M\) as well as \(V\subset S^{n+1}\) such that for all \(i=1,\ldots ,l\)

  1. (a)

    \(U_i\) and V are contractible,

  2. (b)

    \(p_i \in U_i\) and \(y_0 \in V\),

  3. (c)

    there are charts \(\psi _i:U_i \rightarrow {\mathbb {R}}^{n+1}\), \(\psi :V \rightarrow {\mathbb {R}}^{n+1}\),

  4. (d)

    \(\omega _i :=\omega |_{U_i}\) is an orientation preserving diffeomorphism onto V.

Since \(\omega \) has odd degree, l has to be an odd number (such maps exists e.g. using the Pontryagin-Thom construction). Furthermore let \(x_0 \in S^n\) be a regular value of \(\eta \) and \(S_0 =\eta ^{-1}(x_0)\). We may assume that \(S_0\) is connected (e.g. see [15, Theorem C]) and \(S_0 \subset V\). Let \(\varphi _0\) be the framing of \(\nu (S_0)\) induced by \(\eta \), then \(0 \ne [S_0,\varphi _0]\in \Omega _1^{\text {fr}}(S^{n+1})\cong \pi _{n+1}(S^n) \cong {\mathbb {Z}}_2\) and therefore by definition we have \({\text {ind}}(S_0,\varphi _0) \ne 0\).

Denote by \(S_i := \omega _i^{-1}(S_0)\) and frame \(\nu (S_i)\) by \(\varphi _0\) and \(d\omega _i\). Then \(C= S_1\cup \ldots \cup S_l\) together with the framings \(\varphi _i\) is a Pontryagin manifold for \(\eta \circ \omega \) to the regular value \(x_0\). Note that \(w(S_i)=0\) for \(i=1,\ldots ,l\), 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 \({\text {ind}}(S_i,\varphi _i) = {\text {ind}}(S_0,\varphi _0)\ne 0\) for all \(i=1,\ldots ,l\) and from that we infer

$$\begin{aligned} \kappa ([\eta \circ \omega ]) = \sum _{i=1}^{l} {\text {ind}}(S_i,\varphi _i) =l \cdot {\text {ind}}(S_0,\varphi _0) \ne 0 \end{aligned}$$

since l is odd, which proves (a).

Part (b) follows directly from part (a). \(\square \)

Corollary 4.2

Suppose M is simply connected, then, up to homotopy, there are exactly two maps \(M \rightarrow S^n\) and one of them is the constant map. The homotopy class of the non-trivial map is represented by \(\eta \circ \omega :M \rightarrow S^n\), see Theorem 4.1.

Finally we would like to show that \(\kappa \) is natural with respect to maps between manifolds which preserve the spin structure

Proposition 4.3

Suppose \(\Phi :M_1 \rightarrow M_2\) 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 \(\Phi \) of \(M_2\). Then for the natural homomorphism \(\Phi ^\# :\pi ^n(M_2) \rightarrow \pi ^n(M_1)\), \(f\mapsto \Phi \circ f\) we have

$$\begin{aligned} \kappa \left( \Phi ^\#(f) \right) = \deg _2 \Phi \cdot \kappa (f). \end{aligned}$$

where \(\deg _2 \Phi \) is the mod 2 degree of \(\Phi \). Therefore using the isomorphism

$$\begin{aligned} \pi ^n(M) \cong H^n(M;{\mathbb {Z}}) \oplus \pi _{n+1}(S^n) \end{aligned}$$

we have

$$\begin{aligned} \Phi ^\# :\pi ^n(M_2) \rightarrow \pi ^n(M_1),\quad (\alpha ,\nu )\mapsto (\Phi ^*(\alpha ),\deg _2 \Phi \cdot \nu ) \end{aligned}$$

Proof

First note that \(\Phi ^\#\) is well-defined on the homotopy class of \(\Phi \). For \(f \in \pi ^n(M_2)\) there is a decomposition \(f=f_\alpha + f_\nu \) with \(\kappa (f_\alpha )=0\), \(f_\alpha ^*(\sigma )=\alpha \) and \(\kappa (f_\nu )=\nu \) as well as \(f_\nu ^*(\sigma )=0\).

Let us show first \(\Phi ^\#(f_\alpha )=f_{\Phi ^*(\alpha )}\). Clearly we have \(\Phi ^\#(f_\alpha )(\sigma )=\Phi ^*(\alpha )\) thus it remains to show \(\kappa (\Phi ^\#(f_\alpha ))=0\). Let \(C_2\) be the preimage of a regular value of \(f_\alpha \) with a normal framing \(\varphi _0\) such that \(\kappa ([C_2,\varphi _0])=0\). Moreover we may choose \(f_\alpha \) such that each framed circle of \((C_2,\varphi _0)\) has index 0. Deform \(\Phi \) to be transversal to \(C_2\), thus \(C_1:=\Phi ^{-1}(C_2)\) 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 \(\Phi \). 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 \(\Phi \) from \(M_2\)) which is also the framing induced by the map \(\Phi \circ f_\alpha \). But this means \(\kappa (\Phi ^\#(f_\alpha ))=0\).

On the other hand we may assume a preimage of a regular point in \(S^n\) under \(f_\nu \) is a contractible circle \(S_2\) in \(M_2\) with normal framing \(\varphi \) such that the index of the framed circle \((S_2,\varphi )\) is \(\nu \in \pi _{n+1}(S^n)\). Then making again \(\Phi \) transverse to \(S_2\) we obtain a normally framed submanifold \((C_1,\varphi )\) such that the index of each framed circle in \(C_1\) has index \(\nu \). As in the proof of Theorem 4.1 the degree of \((C_1,\varphi )\) is just \(\deg _2\Phi \cdot \nu \). Therefore \(\Phi ^\#(f_\nu ) = f_{\deg _2\Phi \cdot \nu }\) and the proposition follows. \(\square \)

Corollary 4.4

Let \(f :M \rightarrow S^n\) and \(x_0 \in S^n\) a regular value. Write \(S_1 \cup \ldots \cup S_k = f^{-1}(x_0)\) such that \(S_i\) is a connected component of \(f^{-1}(x_0)\) and denote by \(\varphi _i\) the induced framing from f. Then the number

$$\begin{aligned} \# \{ i : \kappa ([S_i,\varphi _i])\ne 0 \} \mod 2 \end{aligned}$$

does not depend on \(x_0\) and is a homotopy invariant. Using the notation of Lemma 3.14we can rewrite the above number as

$$\begin{aligned} \# \{ i : \overline{[A_i]}\ne 0 \} \mod 2 \end{aligned}$$

where \([A_i]\) is the homotopy class of matrices corresponding to \([S_i,\varphi _i]\).

Application to vector bundles

In this section \(\pi :E \rightarrow M\) should denote an oriented vector bundle of rank n endowed with a spin structure. Let \(s :M \rightarrow E\) 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 \(ds :TM \rightarrow TE\) restricted to \(\nu (C)\) is an isomorphism of the vector bundles \(\nu (C) \rightarrow E|_C\). Since E possesses a spin structure, by Lemma 3.1\(E|_C\) has a framing and with ds this endows \(\nu (C)\) with the framing \(\varphi \) of \(E|_C\). Note that the homology class \([C] \in H_1(M;{\mathbb {Z}})\) is the Poincaré dual of the Euler class of E.

Proposition 5.1

The class \([C,\varphi ] \in \Omega _1^{\text {fr}}(M)\) does not depend on the section s.

Proof

Let \(s' :M \rightarrow E\) be another transversal section and denote the corresponding normally framed zero locus by \((C',\varphi ')\). Let \(s^*:M \times I \rightarrow \text {pr}^*(E)\) be a section of \(\text {pr}^*(E) \rightarrow M \times I\) (where \(\text {pr} :M \times I \rightarrow M\)) such that \(s^*|_{M \times 0} = s\) and \(s^*|_{M \times 1} = s'\). We may deform \(s^*\) to a section \({\hat{s}}\) which is transverse to the zero section of \(\text {pr}^*(E) \rightarrow M\times I\) and agrees with s and \(s'\) on the boundary of \(M \times I\). The zero locus of \({\hat{s}}\), call it \(\Sigma \subset M \times I\) is a bordism between C and \(C'\) by construction. Moreover by Lemma 3.1\(T(M\times I)|_\Sigma \) inherits a framing from the spin structure of M as well as \(\nu (\Sigma )\) from \(d{\hat{s}}\) and the spin structure of \(\text {pr}^*(E)|_\Sigma \). Thus \(\Sigma \) is a normally framed bordism between \((C,\varphi )\) and \((C',\varphi ')\). \(\square \)

Definition 5.2

The bordism class \([C,\varphi ] \in \Omega _1^{\text {fr}}(M)\) constructed above is called the framed divisor of \(E\rightarrow M\). Furthermore we define the degree \(\kappa (E)\) of E as \(\kappa ([C,\varphi ])\)

For \([C,\varphi ]\in \Omega _1^{\text {fr}}(M)\) we denoted by \(w(C) \in H^{n}(M;{\mathbb {Z}}_2)\) the Poincaré dual of the \({\mathbb {Z}}_2\) fundamental class \([C] \in H_1(M;{\mathbb {Z}}_2)\). If \([C,\varphi ]\) is the framed divisor of \(E \rightarrow M\) then w(C) is the n-th Stiefel-Whitney class \(w_{n}(E)\) (the zero locus of a generic section of E represents the Poincaré dual of the Euler class of E, cf. [4, Proposition 12.8]. Therefore, since n-th Stiefel-Whitney class of E is the Euler class modulo 2, we have that [C] is Poincaré dual to \(w_n(E)\))). Moreover if one changes the spin structure of E by an element \(\alpha \in H^1(M;{\mathbb {Z}}_2)\) then for the framed divisor it is the same as keeping the spin structure of E and changing that of M by \(\alpha \). Hence if \(w_{n}(E)=0\) then the degree of E does not depend on the spin structure (see Lemma 3.1 and Proposition 3.11).

Proposition 5.3

If \(w_n(E)=0\) then the framed divisor is independent of the spin structures on M and E.

For the next theorem we will need a technical Lemma. Let \(D^{m}\) denote the closed unit ball in \({\mathbb {R}}^m\) and consider a smooth map \(f :D^{n+k+1} \rightarrow {\mathbb {R}}^{n+1}\). Assume that \(0 \in {\mathbb {R}}^{n+1}\) is a regular value for f and \(\Sigma _f^k := f^{-1}(0)\) does not intersect the boundary of \(D^{n+k+1}\). Denote by \(\varphi _f\) the induced framing on \(\nu (\Sigma _f^k)\). Since \(\Sigma _f^k\) is a submanifold of \({\mathbb {R}}^{n+k+1}\) the trivialization \(\varphi _f\) defines a stable tangential framing of \(\Sigma _f^k\) thus the pair \((\Sigma _f^k,\varphi _f)\) defines an element in \(\Omega _k^{\text {fr}}\). On the other side, consider

$$\begin{aligned} g :S^{n+k}= \partial D^{n+k+1} \rightarrow S^n,\quad g(x):=\frac{f(x)}{|f(x)|} \end{aligned}$$

and choose a regular value \(y \in S^{n}\). Denote by \((\Sigma _g^k,\varphi _g)\) the induced stably framed manifold.

Lemma 5.4

With the notation above we have that \((\Sigma _f^k,\varphi _f)\) and \((\Sigma _g^k,\varphi _g)\) are stably framed bordant, thus they define the same element in \(\Omega _k^{\text {fr}}\).

Proof

There is an \(\varepsilon >0\) such that the closed ball \(D_\varepsilon \) centered in \(0 \in {\mathbb {R}}^{n+1}\) with radius \(\varepsilon \) contains only regular values of f. The preimage of \(D_\varepsilon \) under f is a disc bundle \(D(\Sigma _f^k)\) of the normal bundle \(\nu (\Sigma _f^k \hookrightarrow {\mathbb {R}}^{n+k+1})\). Denote by \(S(\Sigma _f^k)\) its sphere bundle. Then \(f|_{S(\Sigma _f^k)}\) has image \(S_\varepsilon = \partial D_\varepsilon \). Thus for \(y' \in S_\varepsilon \), \(\Sigma _{y'} = \left( f|_{S(\Sigma _f^k)} \right) ^{-1}(y')\) lies completely in \(S(\Sigma _f^k)\). Moreover the Pontryagin manifold \((\Sigma _{y'},\varphi _{y'})\) is framed bordant to \((\Sigma _f^k,\varphi _f)\). Thus we would like to show that \((\Sigma _{y'},\varphi _{y'})\) represents the same element in \(\Omega _{k}^{\text {fr}}\) as \((\Sigma _g^k,\varphi _g)\). Since the normal bundle of \(S(\Sigma _f^k)\) is trivial the framing \(\varphi _{y'}\) induces a framing \(\varphi _{y'}'\) on \(\nu (\Sigma _{y'}\hookrightarrow S(\Sigma _{f}^k))\) such that \((\Sigma _{y'},\varphi _{y'})\) is stably framed bordant to \((\Sigma _{y'},\varphi _{y'}')\). But the latter normally framed manifold is the Pontryagin manifold to the map \(f|_{S(\Sigma _f^k)} :S(\Sigma _f^k) \rightarrow S_\varepsilon \) at the point \(y ' \in S_\varepsilon \).

Let N be the complement of the interior of \(D(\Sigma _f^k)\) in \(D^{n+k+1}\). Then N is a framed cobordism between \(S^{n+k}=\partial D^{n+k+1}\) and \(S(\Sigma _f^k)\). The restriction of the map

$$\begin{aligned} F :N \rightarrow S^n,\quad F(x):=\frac{f(x)}{|f(x)|} \end{aligned}$$

to \(S^{n+k}\) is equal to g and F restricted to \(S(\Sigma ^k)\) is equal to \(\varepsilon ^{-1}{\hat{f}}\). Hence F defines a framed bordism between \((\Sigma _g^k,\varphi _g)\) and \((\Sigma _{y'},\varphi '_{y'})\) which proves the lemma. \(\square \)

Theorem 5.5

Let \(E \rightarrow M\) 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 \(\kappa (E)=0\).

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 \(\kappa (E)=0\).

Assume now that \(e(E)=0\) and \(\kappa (E)=0\). Consider the fibration

$$\begin{aligned} S^{n-1} \longrightarrow B \text {SO}(n-1) \longrightarrow B \text {SO}(n). \end{aligned}$$

where \(B \text {SO}(k)\) denotes the classifying space to the special orthogonal group \(\text {SO}(k)\). Consider the classifying map \(g :M \rightarrow B \text {SO}(n)\) for \(E \rightarrow M\). There exists a nowhere vanishing section if and only if there is a lift \({\hat{g}}:M \rightarrow B \text {SO}(n-1)\) of g up to homotopy.

First we put a CW-structure on M (e.g. induced by a Morse function) then over the \((n-1)\)-skeleton of M there exists such a lift \({\hat{g}}\) of g. The obstruction to extend the lift over the n-skeleton lies in \(H^n(M;\pi _{n-1}(S^{n-1}))=H^n(M;{\mathbb {Z}})\) which is given by the Euler class e(E). Since this is assumed to be zero \({\hat{g}}\) extends over the n-skeleton of M. The obstruction to extend \({\hat{g}}\) over the top cell of M lies in \(H^{n+1}(M;\pi _{n}(S^{n-1}))\cong \pi _{n}(S^{n-1})\cong {\mathbb {Z}}_2\). Let \(e_{n+1}\) be the top cell of M and \(\psi :\partial e_{n+1}\cong S^{n} \rightarrow M\) the corresponding attaching map. The bundle \(E|_{e_{n+1}}\) is canonical isomorphic to \(e_{n+1} \times {\mathbb {R}}^n\). Let \(\sigma :M \rightarrow E\) 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

$$\begin{aligned} g :\partial e_{n+1}\cong S^n \rightarrow S^{n-1},\quad g(x):=\frac{\sigma \circ \psi (x)}{|\sigma \circ \psi (x)|} \end{aligned}$$

(where the norm is taken with respect to a Euclidean bundle metric on E). The homotopy class of g in \(\pi _{n}(S^{n-1})\) is the obstruction to extend a no where vanishing section over the n-skeleton to the \((n+1)\)-skeleton of M. Since \(\pi _n(S^{n-1})\) is isomorphic to the stable homotopy group \(\pi _1^S\) we consider the homotopy class of g as an element therein.

From Lemma 5.4 we infer that \([g] \in \pi _1^S\cong \Omega _1^{\text {fr}}\) is equal to the framed divisor \(\kappa (E)\) of E defined by \(\sigma \), thus E admits a nowhere vanishing section in case \(e(E)=0\) and \(\kappa (E)=0\). \(\square \)

Example 5.6

As an application of our theory we will reprove the following fact due to Whitehead [23] and Eckmann [6]: The number of linear independent vector fields on \(S^{4k+1}\) is equal to 1 (see also [1] and in [22]).

Denote by \(\langle \cdot , \cdot \rangle \) the standard Euclidean product in \({\mathbb {R}}^{4k+2}\). The vector field

$$\begin{aligned} v :{\mathbb {R}}^{4k+2}\rightarrow {\mathbb {R}}^{4k+2},\quad v(x_1,x_2,\ldots ,x_{4k+2})=(-x_2,x_1,\ldots ,-x_{4k+2},x_{4k+1}) \end{aligned}$$

defines a nowhere vanishing vector field on \(S^{4k+1}\) since \(\langle v(x), x \rangle =0\) for \(x \in S^{4k+1}\). Let E be the subbundle of \(TS^{4k+1}\) 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 EFootnote 2. Since the Euler class of E vanishes, it suffices to show that \(\kappa (E)\) 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

$$\begin{aligned} w :{\mathbb {R}}^{4k+2} \rightarrow {\mathbb {R}}^{4k+2},\quad w(x) = (0,0,-x_5,x_6,x_3,-x_4,-x_9,x_{10},x_7,-x_8,\ldots ) \end{aligned}$$

Since \(\langle w(x), x \rangle = \langle w(x), v(x) \rangle =0\) 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

$$\begin{aligned} S=\{(x_1,x_2,0,\ldots ,0) \in S^{4k+1} : x_1^2+x_2^2 =1\}. \end{aligned}$$

In Example 3.10 we saw that \(TS^{4k+1}|_S\) 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 \(\tau _1,\ldots ,\tau _n\) be a trivialization of \(E|_S\), then, since the spin structure on E is induced by \(TS^{4k+1}\) and v, the map \(S \rightarrow \mathrm {SO}(4k+2)\), \(x\mapsto (x,v(x),\tau _1(x),\ldots ,\tau _n(x))\) 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 \(\kappa (E)=1\) and therefore E does not admit a nowhere vanishing section from Theorem 5.5.

Remark 5.7

In [7, Theorem 1.6] the authors show that for any n-dimensional CW-complex of dimension X and any k-dimensional integral cohomology class \(a \in H^k(X;{\mathbb {Z}})\) there exists an oriented vector bundle over X whose Euler class equals \(2 \cdot N(n,k) \cdot a\).

Suppose \(\dim X = 2k+1\). By Steenrod’s exact sequence (ST) it follows that the Hurewicz map \(\pi ^n(X) \rightarrow H^n(X;{\mathbb {Z}})\) is surjective. Then for every \(a \in H^n(X;{\mathbb {Z}})\) there is a map \(f_a \in \pi ^n(X)\) such that \(f_a^*(\sigma )=a\), where \(\sigma \in H^n(S;{\mathbb {Z}})\) denotes the generator such that \(2\sigma \) equals to the Euler class of the tangent bundle \(TS^n\) of \(S^n\). Clearly the vector bundle \(f_a^*(TS^n)\) has Euler class \(2 \cdot a\) and therefore \(N(2k,2k+1)=1\) in the notation of [7].

Note that any vector bundle over \(S^n\) for \(n\ne 2,4,8\) has an Euler class divisible by 2, cf. [2, 13]. In the cases \(n=2,4,8\) there are real vector bundles whose Euler class is a generator of \(H^n(S^n;{\mathbb {Z}})\), namely the associated bundles to the Hopf fibrations \(S^{2n-1} \rightarrow S^n\). We deduce

Proposition 5.8

Suppose \(n=4\) or \(n=8\) and let M be a \((n+1)\)-dimensional closed spin manifold. Denote by \(\mathrm {Vect}_n(M)\) the set oriented vector bundles over M of rank n up to isomorphism. Let \(E_0 \rightarrow S^{n}\) denote the oriented rank n vector bundle such that the Euler class of \(E_0\) is a generator of \(H^n(S^n;{\mathbb {Z}})\). Then the map

$$\begin{aligned} \pi ^n(M) \rightarrow \mathrm {Vect}_n(M), \quad f\mapsto f^*(E_0) \end{aligned}$$

is injective.

Proof

We consider \(f_1,f_2 \in \pi ^n(M)\) such that \(E_1:=f_1^*(E_0)\cong f_2^*(E_0)=:E_2\) since they represent the Euler class the respective bundles. This implies \(f_1^*(\sigma )=f_2^*(\sigma )\) for a generator in \(H^n(S^n;{\mathbb {Z}})\). Thus it remains to show that \(\kappa (f_1) = \kappa (f_2)\). Let \(x_i \in S^n\) be a regular value for \(f_i\) for \(i=1,2\). There is a section \(\sigma _{0,i} :S^n \rightarrow E_0\) 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 \(\sigma _{0,i}\) can only exist since if the Euler class is a generator, since the index of transverse sections is always \(\pm 1\)). Then \(\sigma _i:=f^*(\sigma _{0,i})\) is a transverse section of \(E_i\). Note that from the Pontryagin-Thom construction we may assume that \(f_i^{-1}(x_i)\) is connected, hence the zero locus of \(\sigma _i\) coincides with \(f_i^{-1}(x_i)\). Moreover the framed divisor of \(E_i\) coincides with the degree of \(f_i\) (cf. Definitions 3.8 and 5.2). Since \(E_1\cong E_2\) we have \(\kappa (E_1) \cong \kappa (E_2)\) by construction of the framed divisor and Proposition 5.1. From \(f_1^{*}(\sigma ) = f_2^{*}(\sigma )\) and \(\kappa (f_1)=\kappa (E_1)=\kappa (E_2)=\kappa (f_2)\) it follows from Theorem 4.1 that \(f_1\) is homotopic to \(f_2\). \(\square \)

Change history

Notes

  1. Take a small embedded closed disc and choose a framing on the circle bounding the disc which cannot be extended over a proper embedded disc in \(M \times I\).

  2. For any pair of orthonormal vector fields \(v_1,v_2\) of \(S^{4k+1}\) one can choose a new pair of orthonormal vector fields which consists of v and a section of E.

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Acknowledgements

The author would like to thank S. Carmeli and M. Miller for helpful discussions in [9].

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Konstantis, P. A counting invariant for maps into spheres and for zero loci of sections of vector bundles. Abh. Math. Semin. Univ. Hambg. 90, 183–199 (2020). https://doi.org/10.1007/s12188-020-00228-6

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