A counting invariant for maps into spheres and for zero loci of sections of vector bundles
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A counting invariant for maps into spheres and for zero loci of sections of vector bundles Panagiotis Konstantis1 Received: 2 March 2020 / Accepted: 30 October 2020 © The Author(s), under exclusive licence to Mathematisches Seminar der Universität Hamburg 2020
Abstract The set of unrestricted homotopy classes [M, Sn ] where M is a closed and connected spin (n + 1)-manifold is called the n-th cohomotopy group 𝜋 n (M) of M. Using homotopy theory it is known that 𝜋 n (M) = H n (M;ℤ) ⊕ ℤ2 . We will provide a geometrical description of the ℤ2 part in 𝜋 n (M) analogous to Pontryagin’s computation of the stable homotopy group 𝜋n+1 (Sn ) . This ℤ2 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 → Sn+1 . Finally we will observe that the zero locus of a section in an oriented rank n vector bundle E → M defines an element in 𝜋 n (M) and it turns out that the ℤ2 part is an invariant of the isomorphism class of E. At the end we show that if the Euler class of E vanishes this ℤ2 invariant is the final obstruction to the existence of a nowhere vanishing section.
1 Introduction Pontryagin computed in [16] the (stable) homotopy group 𝜋n+1 (Sn ) ( n ≥ 3 ) using differential topology. Let us describe briefly his construction, since this paper will generalize his idea. He showed that 𝜋n+1 (Sn ) is isomorphic to the bordism group of closed 1-dimensional submanifolds of ℝn+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 Ωfr1 (ℝn+1 ) . Let (C, 𝜑) be a representative of an element of Ωfr1 (ℝn+1 ) , i.e. C is a union of embedded circles in ℝn+1 and there are maps 𝜑1 , … , 𝜑n ∶ C → ℝn+1 such that (𝜑1 (x), … , 𝜑n (x)) is a basis of 𝜈(C)x for every x ∈ C . Let 𝜑n+1 be a trivialization of the tangent bundle of C such that (𝜑1 (x), … , 𝜑n+1 (x)) is a positive oriented basis of ℝn+1 for every x ∈ C . Without loss of generality we may assume that 𝜑1 , … , 𝜑n+1 is pointwise an orthonormal basis. If (e1 , … , en+1 )
Communicated by Vicente Cortés. * Panagiotis Konstantis [email protected]‑koeln.de 1
Department Mathematik/Informatik, Abteilung Mathematik, Universität zu Köln, Weyertal 86‑90, 50931 Köln, Germany
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denotes the standard basis of ℝn+1 , then consider the map A = (aij ) ∶ C → SO(n + 1) such that
𝜑i (x) =
n+1 ∑
aij (x)ej
j=1
, then Pontryagin defines [16, for x ∈ C . Let 𝜋1 (SO(n + 1)) be identified with ℤ2 Theorem 20]
𝛿(C, 𝜑) ∶= [A] + (n(C)
mod 2)
where [A] denotes the homotopy class of A in 𝜋1 (SO(n + 1)) and n(C) is the number of connected components of C. He showed that 𝛿 is well-defined on Ωfr1 (ℝn+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 Sn but rather of a cohomotopy group of Sn+1 . If X is a CW space then the cohomotopy set of X is defined as the set of (unrestricted
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