Embeddings of finite-dimensional compacta in Euclidean spaces

Abstract

If $g$ is a map from a space $X$ into ${\mathbb{R}}^m$ and $q$ is an integer, let $B_{q,d,m}(g)$ be the set of all lines ${\Pi }^d\subset {\mathbb{R}}^m$ such that $|g^{-1}({\Pi }^d)|\ge q$. Let also $\mathcal{H}(q,d,m,k)$ denote the maps $g:X\to {\mathbb{R}}^m$ such that $\dim B_{q,d,m}(g)\le k$. We prove that for any $n$-dimensional metric compactum $X$ each of the sets $\mathcal{H}(3,1,m,3n+1-m)$ and $\mathcal{H}(2,1,m,2n)$ is dense and $G_{\delta }$ in the function space $C(X,{\mathbb{R}}^m)$ provided $m\ge 2n+1$ (in this case $\mathcal{H}(3,1,m,3n+1-m)$ and $\mathcal{H}(2,1,m,2n)$ can consist of embeddings). The same is true for the sets $\mathcal{H}(1,d,m,n+d(m-d))\subset C(X,{\mathbb{R}}^m)$ if $m\ge n+d$, and $\mathcal{H}(4,1,3,0)\subset C(X,{\mathbb{R}}^3)$ if $\dim X\le 1$.

15 pages