The local defect index up to finite ambiguity

Stefan Bechtluft-Sachs

Marco Hien

Abstract: The rational Hopf invariant of the primary obstruction to a nowhere vanishing section in a vectorbundle is computed in terms of characteristic classes. This leads to a local rigidity result on the local defect index of a map.

We consider a q-dimensional oriented vector bundle E over a cell complex Δ and a section σ of the sphere bundle SE over the p-skeleton of Δ. Let c(σ) ∈ H^{p+1}(Δ; π_{p}(S^{q−1})) denote the primary obstruction to extending σ|_{Δ^{p−1}} to the p+1 skeleton. It is well known that the coefficient group is infinite only if p = q − 1 or p = 4l − 1, q = 2l + 1. In the first case the obstruction is the Euler class of E. In the second case, we compute the image of this obstruction under the rational Hopf invariant h^{ℚ}, i.e. under the composite h^{ℚ} : π_{4l−1}(S^{2l}) ⟶^{h} ℤ ↪ ℚ, where h is the Hopf invariant.

Theorem 1 Let E be an oriented real vector bundle dimension 2l + 1, l ≥ 1 over a 4l-dimensional cell complex Δ. Let σ : Δ^{4l−1} → SE be a section of the sphere bundle SE over the 4l − 2 skeleton of Δ which extends to a section over the (4l − 1)-skeleton. Then the image under the rational Hopf invariant of the primary obstruction c(E, σ) ∈ H^{4l}(Δ; ℤ) to extending σ to Δ is given by

p_{l}(E) − e(σ)^{2} = 4 (−1)^{l} h_{∗}^{ℚ}c(E, σ) ∈ H^{4l}(Δ; ℚ)

where e(σ) ∈ H^{2l}(Δ; ℚ) restricts to the Euler class of the the complementary bundle of σ over Δ^{4l−2}.

Consider a regularly defect cross section σ in a q-dimensional vector bundle E over a manifold M i.e. σ is defined only outside some closed submanifold Δ ⊂ M. The local defect index (at a point x ∈ Δ) is ι(σ) : SN_{x} → SE_{x} where N is the normal bundle of Δ in M, see [3, 4, 2] for these notions in ordered media with defects. Let p be the dimension of N and assume N and E oriented over Δ. Then we may identify ι(σ) ∈ π_{p−1}(S^{q−1}) as the primary obstruction to extending the section σ on SN to DN. If p = q we have that e(E) = ι(σ)e(N). In the case p = 4l, q = 2l + 1 we can apply the previous theorem to the 4l-skeleton of the relative cell complex (DN, SN) and get

4 (−1)^{4l} h^{ℚ}(ι(σ))e(N) = p_{l}(E) − e^{2} ∈ H^{4l}(Δ, ℚ)

where e ∈ H^{2l}(Δ) ≅ H^{2l}(SN) is the Euler class of the complementary bundle of σ on SN. Thus ι(σ) is determined up to finite ambiguity by e (and the bundles N and E) if the Euler class of N is not torsion.

Proof of the Theorem: Consider the Postnikov decomposition of the map BSO(2l) → BSO(2l + 1).

BSO(2l) ⟶^{q} B_{2l−1} → ⋯ → B_{2} → B_{1} → B_{0} = BSO(2l + 1)

where the fibre of the map B_{i+1} → B_{i} is an Eilenberg MacLane space K(π_{2l+i}(S^{2l}), 2l + i). Let c ∈ H^{4l}(B_{2l−1}; π_{4l−1}(S^{2l})) be the primary obstruction to splitting q. The section σ gives rise to a lift of the classifying map ϵ of E to a map ϵ_{σ} : Δ → B_{2l−1} and we have c(E, σ) = ϵ_{σ}^{∗}c.

Clearly q^{∗}c = 0. For i = 1 … 2l − 2 the groups π_{2l+i}(S^{2l}) are finite and the Eilenberg MacLane spaces K(π_{2l+i}(S^{2l}), 2l + i) are rationally acyclic. Hence the composition B_{2l−1} → ⋯ → B_{2} → B_{1} induces an isomorphism in rational cohomology. From the Serre spectral sequence and the fact that H^{∗}(K(ℤ, 2l); ℚ) = ℚ[x], where x ∈ H^{2l}(K(ℤ, 2l); ℚ), we get that H^{4l}(B_{2l−1}; ℚ) is spanned by monomials in the pull back of the universal Pontrijagin classes p_{i} ∈ H(BSO(2l + 1); ℚ) and an element y ∈ H^{2l}(B_{2l−1}; ℚ) whose restriction to the fibre is x and with q^{∗}y = e, where e ∈ H^{2l}(BSO(2l); ℚ) denotes the universal Euler class. Since the only relation among the characteristic classes in H^{∗}(BSO(2l); ℚ) is e^{2} = p_{l}, we infer that the kernel of q^{∗} in H^{4l} is spanned by p_{l} − y^{2}. Therefore we find α ∈ ℚ, such that αc = p_{l} − y^{2} and

αc(E, σ) = αϵ_{σ}^{∗}c = ϵ_{σ}^{∗}(p_{l} − y^{2}) = p_{l}(E) − e^{2} .

In order to determine the value of α it suffices to consider the special case of a real (2l+1)-dimensional vector bundle E over Δ = S^{4l}. Let η : S^{4l−1} → S^{2l} be the Whitehead product [ι_{2l}, ι_{2l}], where ι_{2l} is the identity on S^{2l}. Its Hopf invariant is h(η) = 2 ([5]). For some integer r the composition of the map ρ : S^{4l−1} → S^{4l−1} of degree r with η lifts to a map S^{4l−1} → SO(2l + 1), which we take as the characteristic map for the vector bundle E. The primary obstruction c(E) in the theorem may be identified with [η ∘ ρ] ∈ π_{4l−1}(S^{2l}). The Pontrijagin class p_{l}(E) can be interpreted similiarly: The complexified bundle E ⊗ ℂ admits a reduction to a SU(2l) bundle E′ and (−1)^{l}p_{l}(E) = c_{2l}(E′) = e(E′) is the same as the degree of the characteristic map of E′. Hence we have to compute the degree of the composite

S^{4l−1} ⟶^{E} SO(2l + 1) ↪ SU(2l + 1) ↩ SU(2l) → S^{4l−1}

This is the same as

S^{4l−1} ⟶^{ϵ} ST ↪^{ζ} L := SU(2l + 1)/SU(2l − 1) ↩ S^{4l−1}

where T = T S^{2l} is the tangent bundle. The inclusion on the right hand is the inclusion of the fibre in the sphere bundle L → S^{4l+1} and induces an isomorphism in H^{4l−1}. We will show, that ϵ and ζ induce multiplication with 4r respectively 2 in H^{4l−1}. Then (−1)^{l}p_{l} = 8r = 4h(η ∘ ρ) and the theorem is proved.

For the assertion about ζ observe that the restriction of the bundle L → S^{4l+1} ⊂ ℂ^{2l+1} to S^{2l} ⊂ ℝ^{2l+1} is the sphere bundle S(T ⊕T), where T is the tangent bundle of S^{2l} and ⊕ denotes the fibre product over S^{2l}. The fibre inclusions S^{4l−1} ↪ S(T ⊕ T) and S^{4l−1} ↪ L induce isomorphisms in H^{4l−1}. Thus we have to compute the map H^{4l−1}(S(T ⊕ T)) → H^{4l−1}(ST). By the multiplicativity of Thom classes, we may decompose this map as

H^{4l−1}(S(T ⊕ T)) \frac{∂}{≅} ⟶ H^{0}(S^{2l}) ⟶^{∪e(T)} H^{2l}(S^{2l}) \frac{∂}{≅} ⟵ H^{4l−1}(ST)

where ∂ denotes the connecting homomorphism in the Gysin sequences of T ⊕ T and T respectively. Since e(T) = 2, ζ induces multiplication by 2.

For the statement about ϵ let r be the order of the finite group π_{4l−2}(S^{2l−1}). Fix a unit vector p ∈ S^{2l−1} ⊂ D^{2l} ⊂ ℝ^{2l} and denote by q : S^{2l−1} → SO(2l) the characteristic map of the tangent bundle of S^{2l}, thus

ST = D^{2l} × S^{2l−1}/ ∼ where (x, v) ∼ (p, q(x)v) for x ∈ ∂D^{2l}  (1)

The map x ↦ q(x)p on S^{2l−1} has degree 2 ([1]). We represent the Whitehead product η = [ι_{2l}, ι_{2l}] by the map

S^{4l−1} = D^{2l} × S^{2l−1} ∪ S^{2l−1} × S^{2l−1} × I ∪ S^{2l−1} × D^{2l} ⟶^{pr_{1}∪∗∪pr_{2}} S^{2l} (2)

where pr_{i} denotes the projection on the ith factor and ∗ is the constant map. In order to lift η ∘ ρ we need to lift the r fold connected sum of (2), which is obtained by glueing (2) in the interior of the cylinders S^{2l−1} × S^{2l−1} × I. We can lift η ∘ ρ on the handles D^{2l} × S^{2l−1} and S^{2l−1} × D^{2l} by mapping D^{2l} × S^{2l−1} ∋ (x, y) ↦ (x, q(y)p) ∈ ST and S^{2l−1} × D^{2l} ∋ (x, y) ↦ (y, q(x)p) ∈ ST. By the identification (1) this determines a map λ_{0} : S^{2l−1} × S^{2l−1} × {0, 1} → S^{2l−1} mapping (x, y, 0) ↦ q(x)q(y)p and (x, y, 1) ↦ q(y)q(x)p. Since the bidegree of λ_{0} on both components is (2, 2), we can extend it to (S^{2l−1} ∨ S^{2l−1}) × I. Furthermore S^{2l−1} × S^{2l−1} × I arises from S^{2l−1} × S^{2l−1} × {0, 1} ∪ S^{2l−1} ∨ S^{2l−1} × I by attaching a cell of dimension 4l − 1. Its attaching map composed with λ_{0} yields a map λ : S^{4l−2} → S^{2l−1}. By assumption, rλ is nullhomotopic. Hence we get an extension on the r-fold connected sum of the cylinders.

Clearly the lift so constructed has degree 4r. Eventually substituting r by a multiple we can further lift ϵ to SO(2l + 1) to realize it by a vector bundle E.

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