Details

Autor(en) / Beteiligte

• Jerrard, R. P.

Titel

On Knaster's Conjecture

Ist Teil von

• Transactions of the American Mathematical Society, 1972-01, Vol.170, p.385-402

Ort / Verlag

American Mathematical Society

Erscheinungsjahr

1972

Link zum Volltext

Quelle

American Mathematical Society Publications

Beschreibungen/Notizen

• Knaster's conjecture is: given a continuous g: Sn→ Emand a set Δ of n - m + 2 distinct points (q1, ⋯, qn - m + 2) in Snthere exists a rotation r: Sn→ Snsuch that g(r(q1)) = g(r(q2)) = ⋯ = g(r(qn - m + 2)). We prove a stronger statement about a smaller class of functions. If f: Sn→ Enwe write f = (f1, f2, ⋯, fn) where fi: Sn→ E1, and put Fi= (f1, ⋯, fi): Sn→ Eiso that Fn= f. The level surface of Fiin Sncontaining x is li(x) = {y ∈ Sn∣ Fi(x) = Fi(y)}. Theorem. Given an (n + 1)-frame$\Delta \subset S^n$and a real-analytic function f: Sn→ Ensuch that each li(x) is either a point or a topological (n - i)-sphere, there exist at least 2n - 1distinct rotations r: Sn→ Snsuch that fi(r(q1)) = ⋯ = fi(r(qn - i + 2)), i = 1, 2 ⋯, n. For each rotation. It follows that for m = 1, 2, ⋯, n, Fm(r(q1)) = Fm(r(q2)) = ⋯ = Fm(r(qn - m + 2)), So that the functions Fm: Sn→ Emsatisfy Knaster's conjecture simultaneously. Given Fi, the definition of f can be completed in many ways by choosing fi + 1, ⋯, fn, each way giving rise to different rotations satisfying the Theorem. A suitable homotopy of f which changes fnslightly will give locally a continuum of rotations r each of which satisfies Knaster's conjecture for Fn - 1. In general these exists an (n - m)-dimensional family of rotations satisfying Knaster's conjecture for Fm.