Roberts' type embeddings and conversion of the transversal Tverberg's theorem

Details Roberts' type embeddings and conversion of the transversal Tverberg's theorem Roberts' type embeddings and conversion of the transversal Tverberg's theorem

2005-03-07

arxiv.org/abs/math/0503131v1

Mathematics

General Topology

21 pages

Scientific paper

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.

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