Dubrovnik VI – Geometric Topology September 30 – October 7, 2007
	ABSTRACTS	pdf-version of this abstract

Vitaly V. Fedorchuk
Moscow State University, Russia

Weakly infinite-dimensional compacta and $C$-compacta via polyhedra and simplicial complexes

Below, all spaces are compact.

Definition 1. Let $K$ be a polyhedron. A mapping $f\colon X\to {\rm Cone} (K)$ is said to be $K$-essential if the mapping $f|_{f^{-1} (K)} \colon f^{-1} (K)\to K$ extends over $X$.

For a class $\mathcal K$ of polyhedra, a space $X$ is called $\mathcal K$-weakly infinite-dimensional (notation: $X\in\mathcal{K}$-$wid$) if for any sequence $f_i\colon X\to {\rm Cone} (K_i)$, $i\in \mathbb N$, $K_i\in\mathcal K$, there exists $n$ such that the mapping $$f_1\Delta \dots \Delta f_n\colon X\to \prod _{i=1}^{n} {\rm Cone} (K_i) = {\rm Cone} (\ast _{i=1}^n K_i)$$ is $\left(\ast _{i=1}^n K_i\right)$-inessential. If $\mathcal K$ consists of one polyhedron $K$ we write $\mathcal{K}$-$wid = K$-$wid$.

Definition 2. Let $\mathcal G$ be a class of finite simplicial complexes. A space $X$ is said to be $\mathcal G$-$C$-space (notation: $X\in\mathcal{G}$-$C$) if for each sequence $u_i = \{ U^i_1,\dots , U^i _{k_i}\}$, $i\in\mathbb N$, of covers of $X$ there exist families $v_i =\{ V^i_1,\dots , V^i _{k_i}\}$ of open subsets of $X$ such that

1. $V^i_j\subset U^i_j$;
2. $N(v_i)\subset G_i\in \mathcal G$, where $N(v)$ denotes the nerve of the family $v$;
3. $\cup _{i=1}^n v_i$ is a cover of $X$ for some $n$.

We investigate classes $\mathcal{K}$-$wid$ and $\mathcal G$-$C$ and their relationships with the classes $wid$ of weakly infinite-dimensional compact spaces and $C$ of compact $C$-spaces. In particular, we show that $$wid\subset\mathcal{K}\mbox{-}wid$$ for an arbitrary $\mathcal K$ and $$C =\mathcal{G}\mbox{-}C$$ for an infinite $\mathcal G$ with ${\rm dim} (\mathcal{G}) <\infty$.

Question. Is it true that $wid = K$-$wid$ for a simply connected $K$ with torsion-free homology groups $H_{*} (K)$?