In this work we study the space of group homomorphisms Hom(Γ,G) from a geometric and simplicial point of view. The case in which the source group is a free abelian group of rank n is studied in more detail since this space can be identified with the space of commuting n-tuples of elements from G. This latter case is of particular interest when the target is a Lie group. The simplicial approach allows us to to construct a family of spaces that filters the classifying space of a group by filtering group theoretical information of the given group. Namely, we use the lower central series of free groups to construct a family of simplicial subspaces of the bar construction of the classifying space of a group. The first layer of this filtration is studied in more detail for transitively commutative (TC) groups.

In this work we study the space of group homomorphisms Hom(Γ,G) from a geometric and simplicial point of view. The case in which the source group is a free abelian group of rank n is studied in more detail since this space can be identified with the space of commuting n-tuples of elements from G. This latter case is of particular interest when the target is a Lie group. The simplicial approach allows us to to construct a family of spaces that filters the classifying space of a group by filtering group theoretical information of the given group. Namely, we use the lower central series of free groups to construct a family of simplicial subspaces of the bar construction of the classifying space of a group. The first layer of this filtration is studied in more detail for transitively commutative (TC) groups.

In this work we study the space of group homomorphisms Hom(Γ,G) from a geometric and simplicial point of view. The case in which the source group is a free abelian group of rank n is studied in more detail since this space can be identified with the space of commuting n-tuples of elements from G. This latter case is of particular interest when the target is a Lie group. The simplicial approach allows us to to construct a family of spaces that filters the classifying space of a group by filtering group theoretical information of the given group. Namely, we use the lower central series of free groups to construct a family of simplicial subspaces of the bar construction of the classifying space of a group. The first layer of this filtration is studied in more detail for transitively commutative (TC) groups. / Science, Faculty of / Mathematics, Department of / Graduate

2023 / Topological properties are considered structural properties of objects because these properties are sustained under continuous deformations of objects. Topological invariance provides stability to representations across changes to objects and viewpoints and are therefore central to object representation. I examined the role of topological properties in the cognitive development of object representations in perception, working memory, and categorization. In Chapter 2, I examined the phenomenological perception of hole shape in 3-8-year-olds (N = 133) by exploiting sound-shape correspondence (bouba/kiki effect). I hypothesized that if children directly perceived hole shape, they would show the sound-shape correspondence effect for holes. As predicted, the study findings were that children showed congruency between object hole shapes and the nonsense labels “bouba” and “kiki” suggesting that children from 3 years of age assign the inner contour of an object with a hole to the hole and not the material surrounding the hole. In Chapter 3, I investigated whether topological properties are maintained similarly to surface features in working memory, and whether attention has a similar influence on the maintenance of these representations, in 24-30-month-old children (N = 43). I hypothesized that topological objects would be remembered better than surface features and that attention cues would support children’s encoding of both feature types. Contrary to the predictions, I found that children maintained topological representations on par with surface features, and attention did support encoding in children’s working memory. In Chapter 4, I examined whether 3-8-year-old children (N = 151) use topological properties to make inferences about objects’ categories using a name generalization task. I predicted that, if so, then topology should compete with or even supersede surface features (i.e., shape or color) as the target of children’s extension of novel nouns to novel objects. As predicted, I found that topological properties competed with both shape and color, suggesting a similar but distinct role for objects’ topological properties and objects’ surface features in children’s kind-based inferences. Taken together, these findings show the critical role of considering topological properties in addition to surface features in achieving a comprehensive understanding of the development of perception, working memory, and object categorization.