Algebraic Rational G-Equivariant Stable Homotopy Theory for Profinite Groups and Extensions of a Torus

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This project lies within algebraic topology, which is the area of mathematics devoted to finding abstract notions of shape and applying algebraic methods to study these notions. The primary objects studied in algebraic topology are spaces, simple examples include the circle, the sphere and the torus (an American doughnut). Indeed, any object in real life represents a space. The combination of geometry and algebra and the ubiquity of spaces has helped algebraic topology to become a fascinating area of mathematics that can apply its powerful techniques to many kinds of problems in a wide variety of other scientific disciplines. Many shapes have symmetries, for example the square can be rotated by ninety-degrees or reflected without changing the shape. These symmetries form what is known as a group, since each symmetry can be undone and any two symmetries can be combined. In general, one fixes an abstract group of symmetries G and considers those spaces which have a set of symmetries which behave like G and only considers those operations which respect these symmetries.It is very hard to perform calculations in equivariant homotopy theory, so we simplify the situation by concentrating on only some of the information and ignoring the rest. One useful piece of information about a space is its Betti numbers. The first Betti number of a shape represents the number of cuts that can be made without dividing the shape into two pieces, so the first Betti number of a circle is one. There are higher Betti numbers which count the number of `holes' of a given dimension in a space. The only non-zero Betti number of the circle is the first. Since we want to study spaces with symmetry, we have to consider more than just the Betti numbers of the space. Let H be some smaller collection of symmetries in G, such that the combination of any two symmetries of H is also in H and such that the inverses of elements of H are also in H (H is called a subgroup of G). Then for a space X, we can consider the collection of all points of X that are unchanged by applying any element of H. We call new shape this the H-fixed point subspace of X. Rational equivariant stable homotopy studies spaces and operations on spaces which preserve symmetries and the Betti numbers of each H-fixed subspace of X, as H varies over all possible subgroups.This combination of adding more structure (the symmetries) and then ignoring all but the Betti numbers makes rational equivariant stable homotopy theory both interesting and usable. The aim of this project is to make this area of mathematics even more usable by making it more algebraic. In the case of a finite group G, rational equivariant stable homotopy theory is completely modelled by an algebraic construction. Thus any space is represented by an object of this algebraic construction and all of the (rational equivariant stable homotopy-theoretic) information about this space is contained in this object. This algebraic model (for rational equivariant stable homotopy theory) is much easier to work with and obtain information from. Currently this method of replacing rational G-equivariant homotopy theory by an algebraic model can only be done for finite groups and products of the circle group. This project is designed to extend this work to more general groups. One of the major complications is that infinite groups have a shape themselves and this must be included in the algebraic model. So this project will begin with two generalisations of the known cases. The first is to extend a product of circle groups (which represent rotations) by adding in a finite group (representing reflections). The second is to take an infinite collection of finite groups and piece them together (to obtain a profinite group).