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

在我读博士的头两年里，我一直致力于更好地理解Tom Dieck关于紧李群G的同伦等变边性MU_G（[tD 70]）的结构。MU_G是泛等变复定向理论，当G是交换紧李群时，MU_G的许多结构定理已被证明。例如，如果A是阿贝尔紧李群：- MU_A是MU上的自由模，并且集中在偶数度上（[Com 96]）; - MU_A带有泛A-等变形式群律（[Hau 19]）。在Schwede（[Sch 20]）最近工作的基础上，我证明了Greenlees和May在[GM 97，猜想1.2]中建立的一个猜想。包含证明[GM 97，猜想1.2]的论文将发表在《几何与拓扑》杂志上。当等变来自一个不一定交换的紧李群G时，这个完备定理导致了等变形式群律的问题。事实上，根据[Gre 01]的大纲定义15.1，如果G是非交换的，则G-等变形式群律应该由完备代数的集合组成。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是一个复定向理论。更具体地说，我们可以选择一个完备G-论域U_G的n维Grassmanian作为U（n）的G-等变分类空间的模型。关于U_G的n维Grassmanian的G-等变E-上同调理论是否是E_G的（导出）完备化的问题仍然是一个未解决的问题，[Hau 19，Prop 5.27]要求理解G × U（n）到图子群的限制的核，在E是K-理论和G是有限的情况下，我们知道J（G，第一章（上述理想的交）是由rv_1的Euler类生成的，其中r是G的正则表示，v_1是重言式U（1）表示。1）在从G × U（2）到G × U（1）的限制同态和Schwede在[Sch 20]中定义的这个限制的分裂同态下，记为S.索赔如下：- J（G，2）=（e（rv_2），S（e（rv_1）利用K-理论中对极大环面的限制是单射的这一事实，我们也用v_2的亚当斯运算和r的外幂来描述S（e（rv_1））。希望这是一个归纳论证的第一步，当G是有限的时，它将给出K-理论中对每个n的理想J（G，n）的描述。建立在这一点上，我将能够猜想一个描述J（G，n）也为等变bordism。[Com96] Gustavo Comezana.复等变协边计算。等变同伦与上同调理论，91：333-352，1996. [GM 97] J.P.C. Greenlees和J.P. May。MU-模谱的局部化和完备化定理。Annals of Mathematics，146（3）：509-544，1997. [Gre01] JPC Greenlees.等变形式群律与复上同调理论. Homology，Homotopy and Applications，3（2）：225-263，2001. [Hau 19] Markus豪斯曼。整体群律与等变边环。arXiv预印本arXiv：1912.07583，2019。[Sch20] Stefan Schwede.全局麦基函子的分裂和等变欧拉类的正则性，2020。arXiv：2006.09435。[tD70] Tammo Tom Dieck. G-流形的边数与积分定理Topology，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.