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

有限群和环面扩张的代数有理G-等变稳定同伦理论

• 批准号: EP/H026681/2
• 负责人: David Barnes
• 金额: $ 6.87万
• 依托单位: Queen's University Belfast
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2013
• 资助国家: 英国
• 起止时间: 2013 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FH026681%2F2
• 关键词: Algebraic; Rational; Equivariant; Stable; Homotopy

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).

这个项目属于代数拓扑学，这是数学领域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。代数拓扑学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）。的确，真实的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普遍存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问题。许多形状具有对称性，例如正方形可以旋转90度或反射而不改变形状。这些对称性形成了所谓的群，因为每个对称性都可以被撤销，任何两个对称性都可以被组合。一般来说，我们固定一个抽象的对称群G，并考虑那些具有一组行为类似于G的对称的空间，只考虑那些尊重这些对称的运算。在等变同伦理论中进行计算是非常困难的，所以我们通过只集中于一些信息而忽略其余信息来简化情况。关于空间的一个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进行的切割次数，因此圆的第一个贝蒂数为1。还有更高的贝蒂数，它计算空间中给定维度的“洞”的数量。唯一不为零的Betti数是第一个。由于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贝蒂数。设H是G中某个较小的对称集合，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）。那么对于一个空间X，我们可以考虑X的所有点的集合，这些点通过应用H的任何元素都是不变的。我们称这个新的形状为X的H-不动点子空间。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X的每个H-固定子空间的对称性和Betti数的空间和空间上的运算。这种增加更多结构（对称性）然后忽略除了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可用。这个项目的目的是使数学的这一领域更加可用，使其更具代数性。在有限群G的情况下，有理等变稳定同伦理论完全由代数结构建模。因此，任何空间都可以用这个代数结构的对象来表示，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（对于有理等变稳定同伦理论）更容易使用和获得信息。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对有限群和圈群的乘积进行。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，无限群有一个形状本身，这必须包括在代数模型。所以这个项目将开始与已知的情况下两个概括。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。第二种方法是取有限群的无限集合并将它们拼接在一起（以获得profinite群）。

项目成果

Model categories for orthogonal calculus

正交微积分的模型类别

• DOI: 10.2140/agt.2013.13.959
• 发表时间: 2013
• 期刊: Algebraic & Geometric Topology
• 影响因子: 0.7
• 作者: Barnes D

Homological Localisation of Model Categories

模型类别的同源定位

• DOI: 10.1007/s10485-013-9340-9
• 发表时间: 2013
• 期刊: Applied Categorical Structures
• 影响因子: 0.6
• 作者: Barnes D

STABLE LEFT AND RIGHT BOUSFIELD LOCALISATIONS

稳定的左右布菲尔德定位

• DOI: 10.1017/s0017089512000882
• 发表时间: 2013
• 期刊: Glasgow Mathematical Journal
• 影响因子: 0.5
• 作者: BARNES D

A monoidal algebraic model for rational SO (2)-spectra

有理 SO (2) 谱的幺半群代数模型

• DOI: 10.1017/s0305004116000219
• 发表时间: 2016
• 期刊: Mathematical Proceedings of the Cambridge Philosophical Society
• 影响因子: 0.8
• 作者: BARNES D

Rational $O(2)$-equivariant spectra

有理$O(2)$-等变谱

• DOI: 10.4310/hha.2017.v19.n1.a12
• 发表时间: 2017
• 期刊: Homology, Homotopy and Applications
• 作者: Barnes D