Constructing Counterexamples in Group Rings and Algebraic Topology

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Groups are the mathematical embodiment of the concept of symmetries of an object, and representation theory studies how these symmetries act on space. The representations of a group are intimately connected to an object known as the group ring. The structure of group rings is fiendishly complicated, but there is one type of group, a so-called torsion-free group, for which the structure is supposed to be simple.On the other hand, topology aims to understand the broad-brush structure of geometric objects, and algebraic topology applies tools from algebra to this problem. Many conjectures in algebraic topology concern the group ring, and can be related to purely algebraic properties of it.From around the 1950s, Irving Kaplansky set out a variety of conjectures in ring theory, and three of these - the idempotent conjecture, the zero-divisor conjecture and the unit conjecture - concern the algebraic structure of group rings of torsion-free groups. There is an intricate web of interdependencies between these conjectures and those of algebraic topology, and it leads to a conjecturally very elegant theory for these groups.However, it seems possible that the theory is not nearly so elegant. The unit conjecture in particular does not have direct topological equivalents. It has also been proved for far fewer classes of groups. Could it be false? Could the other two conjectures also be false?This project will study the three conjectures, looking for counterexamples rather than to prove the result for a class of groups. It aims to prove indeed that these groups are more complicated than previously thought, and perhaps the theory needs to be reworked.