On van Kampen-Flores, Conway-Gordon-Sachs and Radon theorems

We exhibit relations between van Kampen-Flores, Conway-Gordon-Sachs and Radon theorems, by presenting direct proofs of some implications between them. The key idea is an interesting relation between the van Kampen and the ConwayGordon-Sachs numbers for restrictions of a map of (d + 2)-simplex to R to the (d+ 1)-face and to the [d/2]-skeleton. We exhibit relations between the following van Kampen-Flores, Conway-GordonSachs and Radon theorems, by presenting direct proofs of some implications between them, see Main Remark 1 below. Thus we obtain alternative proofs of some of these results assuming another. Direct proofs of the implications (1) below were apparently not published before. Such proofs are based on interesting properties of the van Kampen and the Conway-Gordon-Sachs numbers, see Lemma 2 below. Denote by ∆N the N -dimensional simplex. Consider the following assertions for each integer d > 0: (V KFd) Van Kampen-Flores Theorem. Let f : ∆d+2 → R d be a general position PL map. If d is even, then there are disjoint (d/2)-faces whose images intersect. If d is odd, then there is a linked pair of images of boundaries of (d+ 1)/2-simplices with the vertices at these points. 1 (V KF d ) ‘Quantitative’ van Kampen-Flores Theorem. Let f : ∆d+2 → R d be a general position PL map. If d is even, then the number of intersection points in R of images of disjoint (d/2)faces, is odd. I.e. the number of points x ∈ R such that x ∈ f(σ) ∩ f(τ) for some disjoint (d/2)-faces σ, τ , is odd. If d is odd, then the number of linked modulo 2 unordered pairs of images of boundaries of (d+ 1)/2-faces with the vertices at these points, is odd. 2 (TRd) Topological Radon Theorem. For each (continuous or PL) map ∆d+1 → R d there are disjoint faces whose images intersect. (TR d ) ‘Quantitative’ Topological Radon Theorem. For each general position PL map f : ∆d+1 → R d the number of intersection points in R of images of disjoint k and I am grateful to S. Avvakumov, R. Karasev, E. Kolpakov, S. Melikhov and M. Skopenkov for useful discussions. Supported in part by RFBR, Grants No. 15-01-06302, the D. Zimin Dynasty foundation, and the Simons-IUM fellowship. Homepage: www.mccme.ru/~skopenko. . 1 This result is due to Conway-Gordon-Sachs for d = 3 and to Lovas-Schrijver-Taniyama for the general case [CG, Sa81, LS, Ta]. Also there are disjoint (d − 1)/2 and (d + 1)/2-faces whose images intersect. This is weaker than (V KFd) and easily follows from (V KFd−1) by link construction. . 2Also the number of intersection points in R of images of disjoint (d− 1)/2 and (d+1)/2-faces, is even (and non-zero, see footnote 1).