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，它是致力于寻找形状的抽象概念并应用代数方法研究这些概念的数学领域。代数拓扑学的主要研究对象是空间，简单的例子包括圆、球面和环面(一种美国甜甜圈)。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处不在，帮助代数拓扑学成为一个迷人的数学领域，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。这些对称形成了所谓的群，因为每个对称都可以撤销，任何两个对称都可以组合。一般来说，人们固定一个抽象的对称群G，考虑那些具有一组行为类似于G的对称性的空间，并且只考虑那些遵守这些对称性的运算，在等变同伦理论中很难进行计算，所以我们只关注其中的一些信息而忽略其余的信息来简化这种情况。关于空间的一个有用信息是它的Betti数。形状的第一个Betti数表示在不将形状分成两部分的情况下可以切割的次数，因此圆的第一个Betti数是1。有更高的Betti数，它计算空间中给定维度的“洞”的数量。圆的唯一非零Betti数是第一个。既然我们想研究具有对称性的空间，我们必须考虑的不仅仅是空间的Betti数。设H是G中一些较小的对称集合，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的子群)。然后对于一个空间X，我们可以通过应用H的任何元素来考虑X中所有不变的点的集合，我们称之为X的H-不动点子空间的新形状。有理等变稳定同伦研究当H在所有可能的子群上变化时保持对称性和每个H-不动子空间的Betti数的空间和空间上的运算。这种增加更多的结构(对称性)然后忽略除Betti数之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数学更具代数性来使这一数学领域变得更有用。在有限群G的情况下，有理等变稳定同伦理论完全由一个代数构造来建模。因此，任何空间都由这种代数结构的对象来表示，并且关于该空间的所有(有理等变稳定同伦理论)信息都包含在该对象中。这个代数模型(对于有理等变稳定同伦理论)更容易使用和获取信息。目前，这种用代数模型代替有理G-等变同伦理论的方法只适用于有限群和圆群的乘积。该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性是无限群本身有一个形状，这必须包括在代数模型中。因此，这个项目将从两个已知案例的概括开始。第一种是通过添加有限群(表示反射)来扩展圆群(表示旋转)的乘积。第二种方法是获取有限群的无限集合，并将它们拼凑在一起(以获得一个有限群)。

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).