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

这个项目属于代数拓扑学，这是一个致力于寻找形状的抽象概念并应用代数方法来研究这些概念的数学领域。代数拓扑研究的主要对象是空间，简单的例子包括圆、球和环面（一个美国甜甜圈）。事实上，现实生活中的任何物体都代表着一个空间。几何和代数的结合以及空间的无所不在帮助代数拓扑成为一个迷人的数学领域，它可以将其强大的技术应用于各种其他科学学科中的许多类型的问题。许多形状都具有对称性，例如正方形可以旋转90度或在不改变形状的情况下反射。这些对称构成了所谓的群，因为每个对称都可以撤消，任何两个对称都可以组合。一般来说，一个人固定了一个抽象的对称群G然后考虑那些有一组像G一样的对称的空间并且只考虑那些遵从这些对称的操作。在等变同伦理论中进行计算是非常困难的，因此我们通过只关注部分信息而忽略其余信息来简化情况。一个关于空间的有用信息是它的贝蒂数。形状的第一个贝蒂数表示在不将形状分成两部分的情况下可以切割的次数，因此圆形的第一个贝蒂数是1。还有更高的贝蒂数，用来计算空间中给定维度的“洞”的数量。圆的唯一非零的贝蒂数是第一个。既然我们想要研究具有对称性的空间，我们必须考虑的不仅仅是空间的贝蒂数。设H是G中对称的一个较小的集合，使得H的任意两个对称的组合也在H中，并且使得H的元素的逆也在H中（H称为G的一个子群）。那么对于一个空间X，我们可以考虑应用H的任何元素而不变的X的所有点的集合。我们把这种新形状称为X的H不动点子空间。当H在所有可能的子群上变化时，理性等变稳定同伦研究保持对称的空间和空间上的运算以及X的每个H不动子空间的贝蒂数。这种增加更多结构（对称性），然后忽略除贝蒂数以外的所有结构的组合，使得有理等变稳定同伦理论既有趣又实用。这个项目的目的是通过使这个数学领域更加代数化，从而使它更加可用。在有限群G的情况下，用一个代数构造完全模拟了有理等变稳定同伦理论。因此，任何空间都由这个代数构造的对象表示，并且关于这个空间的所有（有理等变稳定同伦论）信息都包含在这个对象中。这种代数模型（用于有理等变稳定同伦理论）更容易处理并从中获得信息。目前这种用代数模型代替有理g -等变同伦理论的方法只能用于有限群和圆群的积。本项目旨在将这项工作扩展到更一般的群体。其中一个主要的复杂问题是无限群本身有一个形状，这必须包含在代数模型中。因此，本项目将从两个已知案例的概括开始。第一种是通过加入有限群（代表反射）来扩展圆群（代表旋转）的积。第二种是取有限群的无限集合，并将它们拼凑在一起（得到一个无限群）。

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).