COMPLETION THEOREMS IN EQUIVARIANT BORDISM AND THE IDEAL J(G; n)

在我博士学位的头两年里，我一直致力于更好地理解Tom Dieck的同伦等变边界定理MU_G([tD70])对于紧李群G的结构。MU_G是泛等变复定向理论，当G是交换紧李群时，许多结构定理都是已知的。例如，如果A是交换紧李群：-MU_A是MU上的自由模，且以偶数次集中([Com96])；-MU_A具有泛A-等变形式群律([Hau19])。在Schwede([Sch20])最近工作的基础上，我证明了Greenlees和May在[GM97，猜想1.2]中建立的一个猜想。包含[GM97，猜想1.2]证明的论文将发表在《几何与拓扑学》杂志上。这个完备性定理引出了当等差来自不一定是交换紧李群G时的等变形式群律问题。实际上，按照[Gre01]的概要定义15.1，如果G是非交换的，则G-等变形式群律应该由一组完备的代数组成。此外，为了更好地理解G-等变形式群律应该是什么，需要研究的主要对象是酉群U(N)的G-等变分类空间的G-等变E-上同调理论，其中E是面向复数的理论。更具体地说，我们可以选择完备G-宇宙U_G的n维草人作为U(N)的G-等变分类空间的模型。关于U_G的n维素数的G-等变E-上同调理论是否是E_G的一个(导出)完备性的问题仍未解决，并遵循[Hau19，Prop 5.27]，这就要求我们理解从G×U(N)到一个图子群的限制的核.在E是K-理论，G是有限的情况下，我们知道J(G，1)(上述理想的交)是由RV_1的欧拉类生成的.其中r是G的正则表示，v_1是重言式U(1)表示，我目前正在研究从G×U(2)到G×U(1)的限制同态下J(G，2)和J(G，1)之间的关系，以及这个限制的分裂同态，这是Schwede在[Sch20]中定义的，我们用S表示。S(e(Rv_1)利用K-理论中对极大环面的限制是内射的这一事实，我们还得到了S(e(Rv_1))关于v_2的Adams运算和r的外幂的一个证明.希望这是一个归纳论证的第一步，它将给出当G是有限时K-理论中每一个n的理想J(G，n)的刻画.在此基础上，我将能够猜测J(G，n)的描述也适用于等变边界论。参考文献[Com96]Gustavo Comezana.复数等变边界论中的计算。等变同伦与上同调理论，91：333-352,1996。访问数/每百万人：Reach for[JP.MU-模谱的局部化和完备化定理。数学年鉴，146(3)：509-544,1997。[Gre01]JPC Greenlees.等变形式群律和面向复数的上同调理论。同调、同伦及其应用，3(2)：225-263,2001。[Hau19]Markus Hausmann。全球群律和等变边界论环。Arxiv预印本arxiv：1912.07583,2019年。[Sch20]斯特凡·施韦德。全局Mackey函子的分裂和等变欧拉类的正则性，2020。ARXIV：2006.09435。书名/作者声明：[by]Tam Dieck.G-流形的边界定理和积分定理。《拓扑学》，9(4)：345-358,1970。

During the first two years of my PhD, I have been focusing in trying to understand better the structure of tom Dieck's homotopical equivariant bordism MU_G ([tD70]) for a compact Lie group G. MU_G is the universal equivariant complex oriented theory and many structure theorems are known when G is an abelian compact Lie group. For example, if A is an abelian compact Lie group: - MU_A is a free module over MU and concentrated in even degrees ([Com96]); - MU_A carries the universal A-equivariant formal group law ([Hau19]). Building upon recent work of Schwede ([Sch20]) I proved a conjecture established by Greenlees and May in [GM97, Conjecture 1.2]. The paper containing the proof of [GM97, Conjecture 1.2] will appear in the journal "Geometry and Topology". This completion theorem led to the question of equivariant formal group law when the equivariance comes from a not necessarily abelian compact Lie group G. In fact, following the outline definition 15.1 of [Gre01] if G is non abelian, a G-equivariant formal group law should consists of a collection of algebras which are complete. Moreover, the main objects to study for better understand what a G-equivariant formal group law should be are the G-equivariant E-cohomology theory of the G-equivariant classifying space of the unitary group U(n), where E is a complex oriented theory. More concretely, we can choose the n-dimensional grassmanians of a complete G-universe U_G as a model for the G-equivariant classifying space of U(n) . The question whether the G-equivariant E-cohomology theory of the n-dimensional grassmanian of U_G is a (derived) completion of E_G is still open and following [Hau19, Prop 5.27] requires the understanding of the kernel of the restriction from G x U(n) to a graph subgroup.In the case E is K-theory and G is finite we know that J(G,1) (the intersection of the ideals above) is generated by the Euler class of rv_1, where r is the regular representation of G and v_1 is the tautological U(1) representation.I am currently investigating the relation between J (G, 2) and J (G, 1) under the restriction homomorphism from G x U(2) to G x U(1) and the splitting homomorphism of this restriction which is defined by Schwede in [Sch20] and we denote by S. The claim is the following: - J(G,2)=(e(rv_2), S(e(rv_1))) Using the fact that the restriction to the maximal torus is injective in K-theory we also have a desctription of S(e(rv_1 )) in terms of Adams operations of v_2 and the exterior power of r. Hopefully, this is the first step of an inductive argument that will give a description of the ideal J(G,n) for every n in K-theory when G is finite. Building up on this I will be able to conjecture a description of J(G,n) also for equivariant bordism. References [Com96] Gustavo Comezana. Calculations in complex equivariant bordism. Equi- variant Homotopy and Cohomology Theory, 91:333-352, 1996. [GM97] J. P. C. Greenlees and J. P. May. Localization and completion theorems for MU-module spectra. Annals of Mathematics, 146(3):509-544, 1997. [Gre01] JPC Greenlees. Equivariant formal group laws and complex oriented co- homology theories. Homology, Homotopy and applications, 3(2):225-263, 2001. [Hau19] Markus Hausmann. Global group laws and equivariant bordism rings. arXiv preprint arXiv:1912.07583, 2019. [Sch20] Stefan Schwede. Splittings of global Mackey functors and regularity of equivariant Euler classes, 2020. arXiv:2006.09435. [tD70] Tammo tom Dieck. Bordism of G-manifolds and integrality theorems. Topology, 9(4):345-358, 1970.