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.
Highlights
Pontryagin computed in [16] the homotopy group n+1(Sn) ( n ≥ 3 ) using differential topology
Let us describe briefly his construction, since this paper will generalize his idea. He showed that n+1(Sn) is isomorphic to the bordism group of closed 1-dimensional submanifolds of Rn+1 furnished with a framing on its normal bundle
More precisely two stably framed manifolds (C0, 0) and (C1, 1) where φi ∶ TCi ⊕ εl → εk+l is an isomorphism are equivalent if there is a bordism Σ between C0 and C1 such that the tangent bundle of Σ possesses a stable framing and the restriction on
Summary
Pontryagin computed in [16] the (stable) homotopy group n+1(Sn) ( n ≥ 3 ) using differential topology. Let us describe briefly his construction, since this paper will generalize his idea He showed that n+1(Sn) is isomorphic to the bordism group of closed 1-dimensional submanifolds of Rn+1 furnished with a framing on its normal bundle 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. We show that 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
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