Continuous maps with the disjoint support property
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Continuous maps with the disjoint support property Snigdha Bharati Choudhury1 · Satya Deo1 Accepted: 2 April 2020 © Akadémiai Kiadó, Budapest, Hungary 2020
Abstract It is well known that for each n ≥ 0, there is a continuous map f : S n → ∂n+1 with the disjoint support property. Since S n and ∂n+1 are homeomorphic, it is natural to ask whether or not there is a homeomorphism h : S n → ∂n+1 with the disjoint support property. In this paper, we prove that there is no such homeomorphism. Further, we also prove the following. Let K n be any triangulation of S 2 having n faces, n ≥ 4. We prove that for triangulation K 6 of S 2 having 6 faces, there exists a continuous map f : S 2 → |K 6 | having the disjoint support property and there is no homeomorphism from S 2 to |K 6 | having the disjoint support property. For each n ≥ 8, we prove that S 2 has at least two non-isomorphic triangulations L n and K n where the first one admits a continuous map f : S 2 → |L n | with the disjoint support property and no homeomorphism with the disjoint support property. For K n , we prove that there is a homeomorphism h : S 2 → |K n | with the disjoint support property. Keywords Disjoint support property · Radon’s theorem · Topological Radon Theorem · Triangulations of sphere Mathematics Subject Classification Primary 52B05; Secondary 54C05
1 Introduction Let S N , N ≥ 0 be an N -sphere and N +1 be an (N + 1)-simplex so that ∂ N +1 is homeomorphic to S N . Let a ∈ N +1 . Then the smallest face of N +1 containing a is called the support [2] of a in N +1 . The support of a may be a vertex, an edge, a triangle etc. More generally, if K is a triangulation of S N , then we can define the support of any point a ∈ |K | as the smallest face of |K | containing a. Let us have the following.
The second author is an honorary scientist of the National Academy of Sciences, India.
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Snigdha Bharati Choudhury [email protected] Satya Deo [email protected]
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Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019, India
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S. B. Choudhury, S. Deo
Definition 1.1 Suppose f : S N → ∂ N +1 is a continuous map. Then we say that f has the disjoint support property if for each pair a, −a of antipodal points of S N , supp f (a) ∩ supp f (−a) = ∅. In other words, the support of the f -images of any pair of antipodal points are disjoint. The Radon Theorem [6,10] says that given any set of d +2 points in Rd , we can decompose the set into two disjoint parts A and B such that conv A ∩ conv B = ∅. Another version of the above theorem is that given any affine map f : d+1 → Rd , there exists two disjoint faces F1 and F2 of d+1 such that f (F1 ) ∩ f (F2 ) = ∅. This theorem was first generalized by Bajmóczy and Bárány [1] in a topological setting as follows: Theorem 1.2 ( [1]) Given a polytope P ⊂ Rd+1 with nonempty interior and a continuous map f : ∂ P → Rd , there exists a pair of opposite faces B and C of P such that f (B)∩ f (C) = ∅. In particular, the topological version of the Radon Theorem says: Theorem 1.
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