Braid groups, mapping class groups, and Torelli groups

This thesis discusses subgroups of mapping class groups of particular surfaces. First, we study the Torelli group, that is, the subgroup of the mapping class group that acts trivially on the first homology. We investigate generators of the Torelli group, and we give an algorithm that factorizes elements of the Torelli group into products of particular generators. Furthermore, we investigate normal closures of powers of standard generators of the mapping class group of a punctured sphere. By using the Jones representation, we prove that in most cases these normal closures have infinite index in the mapping class group. We prove a similar result for the hyperelliptic mapping class group, that is, the group that consists of mapping classes that commute with a fixed hyperelliptic involution. As a corollary, we recover an older theorem of Coxeter (with 2 exceptional cases), which states that the normal closure of the m-th power of standard generators of the braid group has infinite index in the braid group. Finally, we study finite index subgroups of braid groups, namely, congruence subgroups of braid groups. We discuss presentations of these groups and we provide a topological interpretation of their generating sets.

Mapping and modulating spatial attention asymmetries in young and older adults

Healthy young adults demonstrate a group-level, systematic preference for stimuli presented in the left side of space relative to the right (‘pseudoneglect’) (Bowers & Heilman, 1980). This results in an overestimation of features such as size, brightness, numerosity and spatial frequency in the left hemispace, probably as a result of right cerebral hemisphere dominance for visuospatial attention. This spatial attention asymmetry is reduced in the healthy older population, and can be shifted entirely into right hemispace under certain conditions. Although this rightward shift has been consistently documented in behavioural experiments, there is very little neuroimaging evidence to explain this effect at a neuroanatomical level. In this thesis, I used behavioural methodology and electroencephalography (EEG) to map spatial attention asymmetries in young and older adults. I then use transcranial direct current stimulation (tDCS) to modulate these spatial biases, with the aim of assessing age-related differences in response to tDCS. In the first of three experiments presented in this thesis, I report in Chapter Two that five different spatial attention tasks provide consistent intra-task measures of spatial bias in young adults across two testing days. There were, however, no inter-task correlations between the five tasks, indicating that pseudoneglect is at least partially driven by task-dependent patterns of neural activity. In Chapter Three, anodal tDCS was applied separately to the left (P5) and right (P6) posterior parietal cortex (PPC) in young and older adults, with an aim to improve the detection of stimuli appearing in the contralateral visual field. There were no age differences in response to tDCS, but there were significant differences depending on baseline performance. Relative to a sham tDCS protocol, tDCS applied to the right PPC resulted in maintained visual detection across both visual fields in adults who were good at the task at baseline. In contrast, left PPC tDCS resulted in reduced detection sensitivity across both visual fields in poor performers. Finally, in Chapter Four, I report a right-hemisphere lateralisation of EEG activity in young adults that was present for long (but not short) landmark task lines. In contrast, older adults demonstrated no lateralised activity for either line length, thus providing novel evidence of an age-related reduction of hemispheric asymmetry in older adults. The results of this thesis provide evidence of a highly complex set of factors that underlie spatial attention asymmetries in healthy young and older adults.