zbMATH — the first resource for mathematics

Geometric proof of the easy part of the Hopf invariant one theorem. (English) Zbl 0941.55011
A geometric proof is given of Adem’s result that if there is a map \(f: S^{2n-1}\to S^n\) with Hopf invariant 1, then \(n\) is a power of 2. It is shown at first, that this result is equivalent to the statement: “If there is a closed \(n-1\) dimensional manifold \(M^{n-1}\) having an immersion \(g\: M^{n-1}\to R^{2n-2}\) with trivial normal bundle and odd number of double points, then \(n\) is a power of 2.” A short proof of this statement is presented at the end of the paper.

55Q25 Hopf invariants
57R42 Immersions in differential topology
Full Text: EuDML
[1] ADAMS J. F.: On the non-existence of elements of Hopf invariant one. Ann. of Math. (2) 72 (1960), 20-104. · Zbl 0096.17404
[2] ADEM J.: The iteration of Steenrod squares in algebraic topology. Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 720-726. · Zbl 0048.17002
[3] HIRSCH M.: Immersions of manifolds. Trans. Amer. Math. Soc. 93 (1959), 242-276. · Zbl 0113.17202
[4] MILLER J. G.: Self-intersections of some immersed manifolds. Trans. Amer. Math. Soc. 136 (1969), 329-338. · Zbl 0186.57401
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.