Roberts’ Type Embeddings and Conversion of the Transversal Tverberg’s Theorem

Here are two of our main results: Theorem 1. Let X be a normal space with dimX = n and m ≥ n + 1. Then the space C∗(X,R m ) of all bounded maps from X into R m equipped with the uniform convergence topology contains a dense Gδ-subset consisting of maps g such that g(X) ∩Π is at most (n+ d −m)-dimensional for every d-dimensional plane Π in R m , where m − n ≤ d ≤ m. Theorem 2. Let X be a metrizable compactum with dimX ≤ n and m ≥ n + 1. Then, C(X,R m ) contains a dense Gδ-subset of maps g such that for any integers t, d, T with 0 ≤ t ≤ d ≤ m − n − 1 and d ≤ T ≤ m and any d-plane Π ⊂ R m parallel to some coordinate planes Π ⊂ Π in R m , the inverse image g(Π) has at most q points, where q = d + 1 − t + n + (n + T − m)(d − t) m − n − d if n ≥ (m − n − T )(d − t) and q = 1 + n m − n − T otherwise. In case m = 2n + 1, the combination of Theorem 1 and the Nöbeling– Pontryagin embedding theorem provides a generalization of a theorem due to Roberts [20]. Theorem 2 extends the following results: the Nöbeling– Pontryagin embedding theorem (d = 0, m = T ≥ 2n + 1); Hurewicz’s theorem [15] about mappings into an Euclidean space with preimages of small cardinality (d = 0, n + 1 ≤ m = T ≤ 2n); Boltyanski’s theorem [6, Theorem 1] about k-regular maps (d = k − 1, t = 0, T = m ≥ nk + n + k) and Goodsell’s theorem [12] about existence of special embeddings (t = 0, T = m). An infinite-dimensional analogue of Theorem 2 is also established. Our results are based on Theorem 1.1 below which is considered as a converse assertion of the transversal Tverberg’s theorem and implies the Berkowitz-Roy theorem [1], [12]. 1991 Mathematics Subject Classification. Primary: 54F45; Secondary: 55M10, 54C65.