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 dim X=n and m\geq 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 G_{\delta}-subset consisting of maps g such that \bar{g(X)}\cap\Pi^d is at most (n+d-m)-dimensional for every d-dimensional plane \Pi^d in R^m, where m-n\leq d\leq m. Theorem 2. Let X be a metrizable compactum with dim X\leq n and m\geq n+1. Then, C(X,R^m) contains a dense G_{\delta}-subset of maps g such that for any integers t,d,T with 0\leq t\leq d\leq m-n-1 and d\leq T\leq m and any d-plane \Pi^d\subset R^m parallel to some coordinate planes \Pi^t\subset\Pi^T in R^m, the inverse image g^{-1}(\Pi^d) has at most q points, where q=d+1-t+\frac{n+(n+T-m)(d-t)}{m-n-d} if n\geq (m-n-T)(d-t) and q=1+\frac{n}{m-n-T} otherwise. In case m=2n+1, the combination of Theorem 1 and the Nobeling--Pontryagin embedding theorem provides a generalization of a theorem due to Roberts. Theorem 2 extends the following results: the N\"{o}beling--Pontryagin embedding theorem (d=0, m=T\geq 2n+1); Hurewicz's theorem about mappings into an Euclidean space with preimages of small cardinality (d=0, n+1\leq m=T\leq 2n); Boltyanski's theorem about k-regular maps (d=k-1, t=0, T=m\geq nk+n+k) and Goodsell's theorem 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.