We study the homomorphism induced on cohomology by the maximal equicontinuous factor map of a tiling space. We will see that in degree one this map is injective and has torsion free cokernel. We show by example, however, that, in degree one, the cohomology of the maximal equicontinuous factor may not be a direct summand of the tiling cohomology.

We prove a general theorem about preservation of the covering dimension $dim$ by certain covariant functors that implies, among others, the following concrete results. If $G$ G is a pathwise connected separable metric...

Let $\tau$ be an uncountable regular cardinal and $G$ a $T_1$ topological group. We prove the following statements: (1) If $\tau$ is homeomorphic to a closed subspace of $G$, $G$ is Abelian, and the order of every non-neutral element of $G$ is greater than $5$ then $\tau \times \tau$ embeds in $G$ as a closed subspace. (2) If $G$ is Abelian, algebraically generated by $\tau \subset G$, and the order of every element does not exceed $3$ then $\tau \times \tau$ is not embeddable in $G$. (3) There exists an Abelian topological group $H$ such that ${\omega }_1$ is homeomorphic to a closed subspace...

We show that there exists an Abelian topological group $G$ such that the operations in $G$ cannot be extended to the Dieudonné completion $\mu G$ of the space $G$ in such a way that $G$ becomes a topological subgroup of the topological group $\mu G$. This provides a complete answer to a question of V.G. Pestov and M.G. Tkačenko, dating back to 1985. We also identify new large classes of topological groups for which such an extension is possible. The technique developed also allows to find many new solutions to the...

The natural quotient map q from the space of based loops in the Hawaiian earring onto the fundamental group provides a naturally occuring example of a quotient map such that q × q fails to be a quotient map. With the quotient topology, this example shows π₁(X,p) can fail to be a topological group if X is locally path connected.