The word problem for some uncountable groups given by countable words

We investigate the fundamental group of Griffiths' space, and the first singular homology group of this space and of the Hawaiian Earring by using (countable) reduced tame words. We prove that two such words represent the same element in the corresponding group if and only if they can be carried to the same tame word by a finite number of word transformations from a given list. This enables us to construct elements with special properties in these groups. By applying this method we prove that the two homology groups contain uncountably many different elements that can be represented by infinite concatenations of countably many commutators of loops. As another application we give a short proof that these homology groups contain the direct sum of 2^{\aleph_0} copies of \mathbb{Q}. Finally, we show that the fundamental group of Griffith's space contains \mathbb{Q}.Comment: 24 pages, 7 figure

Embeddability of multiple cones

The main result of this paper is that if $X$ is a Peano continuum such that its $n$-th cone $C^n(X)$ embeds into \RR^{n+2} then $X$ embeds into $S^2$. This solves a problem proposed by W. Rosicki.Comment: Some minor corrections were mad

New Constraints on the Escape of Ionizing Photons From Starburst Galaxies Using Ionization-Parameter Mapping

The fate of ionizing radiation in starburst galaxies is key to understanding cosmic reionization. However, the galactic parameters on which the escape fraction of ionizing radiation depend are not well understood. Ionization-parameter mapping provides a simple, yet effective, way to study the radiative transfer in starburst galaxies. We obtain emission-line ratio maps of [SIII]/[SII] for six, nearby, dwarf starbursts: NGC 178, NGC 1482, NGC 1705, NGC 3125, NGC 7126, and He 2-10. The narrow-band images are obtained with the Maryland-Magellan Tunable Filter at Las Campanas Observatory. Using these data, we previously reported the discovery of an optically thin ionization cone in NGC 5253, and here we also discover a similar ionization cone in NGC 3125. This latter cone has an opening angle of 40+/-5 degrees (0.4 ster), indicating that the passageways through which ionizing radiation may travel correspond to a small solid angle. Additionally, there are three sample galaxies that have winds and/or superbubble activity, which should be conducive to escaping radiation, yet they are optically thick. These results support the scenario that an orientation bias limits our ability to directly detect escaping Lyman continuum in many starburst galaxies. A comparison of the star-formation properties and histories of the optically thin and thick galaxies is consistent with the model that high escape fractions are limited to galaxies that are old enough (> 3 Myr) for mechanical feedback to have cleared optically thin passageways in the ISM, but young enough (< 5 Myr) that the ionizing stars are still present.Comment: Accepted for publication in Ap

On Snake cones, Alternating cones and related constructions

We show that the Snake on a square $SC(S^1)$ is homotopy equivalent to the space $AC(S^1)$ which was investigated in the previous work by Eda, Karimov and Repov\vs. We also introduce related constructions $CSC(-)$ and $CAC(-)$ and investigate homotopical differences between these four constructions. Finally, we explicitly describe the second homology group of the Hawaiian tori wedge

Everyone Take a Knee and Listen Up! Examining Student-Athlete Protests During the National Anthem

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