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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
##### Keywords:
framed manifold; immersion; double point
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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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