General position properties in fiberwise geometric topology
Tom 491 / 2013
Streszczenie
General position properties play a crucial role in geometric and infinite-dimensional topologies. Often such properties provide convenient tools for establishing various universality results. One of well-known general position properties is DD$^n$, the property of disjoint $n$-cells. Each Polish ${\rm LC}^{n-1}$-space $X$ possessing DD$^n$ contains a topological copy of each $n$-dimensional compact metric space. This fact implies, in particular, the classical Lefschetz–Menger–Nöbeling–Pontryagin–Tolstova embedding theorem which says that any $n$-dimensional compact metric space embeds into the $(2n+1)$-dimensional Euclidean space $\mathbb R^{2n+1}$. A parametric version of this result was recently proved by B. Pasynkov: any $n$-dimensional map $p:K\to M$ between metrizable compacta with $\dim M=m$ embeds into the projection $\mathrm{pr}_M:M\times\mathbb R^{2n+1+m}\to M$ in the sense that there is an embedding $e:K\to M\times \mathbb R^{2n+1+m}$ with $\mathrm{pr}_M\circ e=p$. This feature of $\mathbb R^{2n+1+m}$ can be derived from the fact that the space $\mathbb R^{2n+1+m}$ satisfies the general position property ${m}\mbox{-}\overline{{\rm DD}\hskip-1pt}\hskip1pt{}^{n}={m}\mbox{-}\overline{{\rm DD}\hskip-1pt}\hskip1pt{}^{\{n,n\}}$, which is a particular case of the 3-parameter general position property ${{m}\mbox{-}\overline{{\rm DD}\hskip-1pt}\hskip1pt{}^{\{n,k\}}}$ introduced and studied in this paper. We shall give convenient “arithmetic” tools for establishing the ${m}\mbox{-}\overline{{\rm DD}\hskip-1pt}\hskip1pt{}^{\{n,k\}}$-property and on this base obtain simple proofs of some classical and recent results on (fiber) embeddings. In particular, the Pasynkov theorem mentioned above, as well as the results of P. Bowers and Y. Sternfeld on embedding into a product of dendrites, follow from our general approach. Moreover, the arithmetic of the ${m}\mbox{-}\overline{{\rm DD}\hskip-1pt}\hskip1pt{}^{\{n,k\}}$-properties established in our paper generalizes some results of W. Mitchell, R. Daverman and D. Halverson.
The paper consists of two parts. In the first part we survey the principal results proved in this paper and discuss their applications and interplay with existing results in this area. The second part contains the proofs of the principal results announced in the first part.
