Narrow Results

In this paper we provide a geometric characterization of those locally compact Hausdorff topological groups which admit a faithful strongly chaotic continuous action on some Hausdorff space.

Let ζ be an n-dimensional complex matrix bundle over a compact metric space X and let A_ζ denote the C*-algebra of sections of this bundle. We determine the rational homotopy type as an H-space of UA_ζ, the group of unitaries of A_ζ. The answer turns out to be independent of the bundle ζ and depends only upon n and the rational cohomology of X. We prove analogous results for the gauge group and the projective gauge group of a principal bundle over a compact metric space X.

The paper examines various properties of $DT$-$k$-subgroup structures and addresses an open problem on the existence of topological group structures on the $n$-dimensional Khalimsky ($K$-, for brevity) topological space and Marcus–Wyse ($M$-, for short) topological plane. In particular, we obtain many types of totally $k$-disconnected or $k$-connected subgroups of a $k$-connected $DT$-$k$-group. Besides, we prove that each of the $n$-dimensional $K$-topological space and the $M$-topological plane cannot be a typical topological group. Unlike an existence of a $DT$-$k$-group structure of $(S{C}_k^{n,l},\ast )$ (see Proposition 4.7), we prove that neither of $(S{C}_K^{n,l},\ast )$ and $(S{C}_M^l,\ast )$ is a topological group, where $S{C}_K^{n,l}$ (respectively, $S{C}_M^l$) is a simple closed $K$- (respectively, $M$-) topological curve with $l$ elements in ${\mathbb{Z}}^n$ (respectively, ${\mathbb{Z}}^2$) and the operation “∗” is a special kind of binary operation for establishing a group structure of each of $S{C}_K^{n,l}$ and $S{C}_M^l$. Finally, given a $DT$-$k$-group structure of $(S{C}_{k_1}^{n_1,l_1}\times S{C}_{k_2}^{n_2,l_2},\ast )$, we find several types of $DT$-$k$-subgroup structures of it (see Theorem 5.7).

This paper is partly a survey on behavior of products in coreflective classes of some topological categories and partly it brings some new results completening those known.