Hereditarily aspherical compacta

Abstract:

The notion of (strongly) hereditarily aspherical compacta introduced by Daverman (1991) is modified. The main results are:

1. If $X\in LC^{1}$ is a hereditarily aspherical compactum, then $X\in$ ANR. In particular, $X$ is strongly hereditarily aspherical.

2. Suppose $f:X\to Y$ is a cell-like map of compacta and $f^{-1}(A)$ is shape aspherical for each closed subset $A$ of $Y$. Then

3. $Y$ is hereditarily shape aspherical,

4. $f$ is a hereditary shape equivalence,

5. $\dim X\ge \dim Y$.

6. Suppose $G$ is a group containing integers. Then the following conditions are equivalent:

7. $\dim X\le 2$ and $\dim _{G}X=1$,

8. $\dim _{G*_{\mathbf {Z}}G}X=1$.

9. Suppose $G$ is a group containing integers. If $\dim X\le 2$ and $\dim _{G}X=1$, then $X$ is hereditarily shape aspherical.

10. Let $X$ be a two-dimensional, locally connected and semilocally simply connected compactum. Then, for any compactum $Y$ \begin{equation*}\dim (X \times Y) = \dim X + \dim Y.\end{equation*}