Hopf Construction Research Articles

Construction of harmonic maps between pseudo-Riemannian spheres and hyperbolic spaces

Defined here is an orthogonal multiplication for vector spaces with indefinite nondegenerate scalar product. This is then used, via the Hopf construction, to obtain harmonic maps between pseudo-Riemannian spheres and hyperbolic spaces. Examples of harmonic maps are constructed using Clifford algebras.

Harmonic maps of spheres and the Hopf construction

On produit des applications harmoniques entre des spheres de grande dimension en utilisant la construction de Hopf. Si p≥6, a tout element du groupe stable Πp dans l'image du J-homomorphisme stable est donnee une representation harmonique

Consequences of the kahn–priddy theorem in homotopy and geometry

Much recent attention has been given to geometric representation of elements of the stable homotopy groups of spheres, π* s A particular example concerns non-singular bilinear maps ℝ m +1 × ℝ n +1 → ℝ m+n +1− p ; on restriction and normalisation these become biskew maps S m × S n ℝ S m+n-p ;. Now the Hopf construction ℋ applied to any map f: S m × S n → S m+n-p yields

Twisted Whitehead Products

By combining the Hopf construction and the Whitehead product map construction we introduce twisted Whitehead products and then extend several well-known properties of Whitehead products to the twisted situation.

Twisted Whitehead products

By combining the Hopf construction and the Whitehead product map construction we introduce twisted Whitehead products and then extend several well-known properties of Whitehead products to the twisted situation.

Non-singular bilinear maps and stable homotopy classes of spheres

A bilinear map ø Ra x Rb → Rc is non-singular if ø (x, y) = 0 implies x = 0 or y = 0. For background information on such maps see (4, 5, 6, 14). If we apply the ‘Hopf construction’ to ø, we get a mapdefined by 2ø(x, y)) for all x ∈ Ra, y ∈ Rb satisfying ∥x∥2 + ∥y∥2 = 1. Homotopically, Jø coincides with the map obtained by first restricting and normalizing ø to , and then applying the standard Hopf construction ((13), p. 112). In any case, one gets an element [Jø] in , which in turn determines a stable homotopy class of spheres {Jø} in the d-stem , where d = a + b − c −1. An element in which equals {Jø} for some non-singular bilinear map ø will be called bilinearly representable. The first purpose of this paper is to prove

A note on a theorem of Ganea, Hilton and Peterson

We observe, of course, that the condition for nil X < 1 is precisely c = 0. Dually, let c': 2 (X X X) -2 (X X X) V2 (X X X) be the cocommutator map. This gives a map c': XXXQ(2XV2X). The Hopf construction then gives a map J(c'): 2(XAX)2Q(2XV2X). Let e: 2Q(2XV2;X)X2X be the map having lI(Nxv x) as its adjoint. Let us denote the cocommutator product 2X-*ZXV2X by c' also. The condition for conil X ? 1 is precisely c' = 0. We prove

A Generalization of a Theorem of Hilton

Let f: A×B → X be a map. Let J(f): ∑(A ∧ B)→ ∑X be the map obtained from f by means of the Hopf construction. Let P(f) denote the space obtained from ∑X by attaching a cone on ∑(A ∧ B) by means of J(f). Let l: ∑X→P(f ) be the inclusion and T(l): X→ΩP(f) the adjoint of l. Let h1,: A1, → A, h2: B1,→ B be maps. Let c:ΩP(f)× ΩP(f) → ΩP(f) be the basic commutator. Then we prove that there exists a map ∑A1 × ∑B1 →P(f) with axes l∑(fi1 h1), l∑(fi2 h2) if and only if , where i1: A→ A × B and i2 B → A × B are the inclusions. This generalizes a result of Hilton. Also, by letting f be an H-space multiplication and h and h the identity maps, we obtain a well known criterion of Stasheff for an H-space to be homotopy-commutative. Finally, appropriate duals of these results are given.

Generalized James product and the Hopf construction

Generalized James product and the Hopf construction

On the Hopf-Toda invariant

0. Introduction. One line of attack on the problem of computing the unstable homotopy groups of spheres is to attempt to construct the elements whose reduced product filtration exceeds 1. The classical Hopf construction associates with a map S' x St1 S of type (oc,,B) an element of itm+t(S'+). The indeterminacy of the construction is the suspension subgroup in the sense that the elements associated with any two maps of the same type differ by a suspension. However, it was proved by I. M. James [8] that the filtration of the element obtained does not exceed 2. In [3] I defined a generalization of the Hopf construction which can yield elements of arbitrarily large filtration. Thus, for example, let a denote the homotopy boundary homomorphism and let Ox E 7rr(Sr'2), fi ;(SL1, eSr r2) be elements(1) such that the Whitehead product [cc,afl] is trivial. Then there exists a map F: S'm X St1 * S-2 of type (, af). Applying the construction we obtain an element C(F) e7E i+t(Sn+ 1) with indeterminacy equal to the subgroup of filtration r 2. As always the problem is to prove nontriviality. To this end we study a homomorphism