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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.

MSC:
55Q25 Hopf invariants
57R42 Immersions in differential topology
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References:
[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
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