[Paper Review] Survey on the Burnside ring of compact Lie groups

Halvard Fausk|ArXiv.org|Jun 15, 2008

Homotopy and Cohomology in Algebraic Topology11 references3 citations

TL;DR

This paper provides a comprehensive survey of the Burnside ring of compact Lie groups, generalizing the classical Burnside ring of finite groups by restricting to orbits with finite Weyl group. It establishes foundational properties, including the mark homomorphism, double coset formulas, and connections to equivariant stable homotopy theory, culminating in the key result that the Euler characteristic map from the Burnside ring to the zeroth equivariant stable homotopy group of the sphere is an isomorphism.

ABSTRACT

The definition and basic properties of the Burnside ring of compact Lie groups are presented, with emphasis on the analogy with the construction of the Burnside ring of finite groups.

Motivation & Objective

• To extend the classical Burnside ring construction from finite groups to compact Lie groups by focusing on orbits with finite Weyl group.
• To clarify the algebraic and topological structure of the Burnside ring in the context of compact Lie groups.
• To establish connections between the Burnside ring and equivariant stable homotopy theory, particularly via the Euler characteristic and transfer maps.
• To provide a systematic overview of key tools such as the mark homomorphism, double coset formulas, and the role of Weyl groups.
• To highlight deep results such as the isomorphism between the Burnside ring and the zeroth equivariant stable homotopy group of the sphere.

Proposed method

• Define the Burnside ring of a compact Lie group as the Grothendieck group completion of finite disjoint unions of transitive $G$-orbits $G/H$ where $H$ has finite Weyl group in $G$.
• Use the double coset formula to define multiplication in the Burnside ring via the decomposition of $G/H \times G/K$ into transitive $G$-orbits.
• Introduce the mark homomorphism $\phi: A(G) \to \prod_{(H)} \mathbb{Z}$, sending a $G$-set to the tuple of fixed point counts $|X^H|$ for each conjugacy class of subgroups $H$
• Leverage Illman's result that $G/H \times G/K$ is a finite $G$-CW-complex to ensure finiteness conditions for orbit type decomposition.
• Establish the Euler characteristic map $\chi: A(G) \to \pi_0^G(S^0)$ via the composition of transfer and collapse maps in equivariant stable homotopy theory.
• Prove the isomorphism $\chi: A(G) \xrightarrow{\sim} \pi_0^G(S^0)$ using the Segal–tom Dieck splitting theorem and the injectivity of the degree homomorphism $d$.

Experimental results

Research questions

• RQ1 How can the Burnside ring construction for finite groups be generalized to compact Lie groups, particularly in terms of orbit types and Weyl groups?
• RQ2 What is the role of the mark homomorphism in encoding fixed point data for $G$-sets under the compact Lie group setting?
• RQ3 How does the Burnside ring relate to the zeroth equivariant stable homotopy group of the sphere, $\pi_0^G(S^0)$?
• RQ4 What conditions ensure that the Euler characteristic map $\chi: A(G) \to \pi_0^G(S^0)$ is an isomorphism?
• RQ5 In what cases does the exponential map from the representation ring to the unit group of the Burnside ring become surjective?

Key findings

• The Euler characteristic map $\chi: A(G) \to \pi_0^G(S^0)$ is an isomorphism, establishing a fundamental link between the Burnside ring and equivariant stable homotopy theory.
• The mark homomorphism $\phi: A(G) \to \prod_{(H)} \mathbb{Z}$ is injective, with $\phi$ factoring through the degree homomorphism $d \circ \chi$, which confirms the algebraic structure of $A(G)$.
• For a compact Lie group $G$ with $|G|$ odd, the element $\frac{u+1}{2}$ lies in $A(G)$ and induces a bijection between idempotent and unit elements in $A(G)$, under specific congruence conditions.
• The composition $A(G) \to R(G;\mathbb{R}) \to A(G)^\times$ is the exponential map, and it is surjective when $G$ is a $2$-group with no subquotients isomorphic to the dihedral group of order 16.
• The canonical map $A(G) \to R(G;k)$ sending $G/H$ to the alternating sum of cohomology representations $H^i(G/H; k)$ is neither injective nor surjective in general, but becomes surjective for $p$-groups when $k = \mathbb{Q}$.
• The Segal conjecture, proved by Carlsson, identifies the stable homotopy classes $[\Sigma^\infty_G EG_+, \Sigma^\infty_G S^0]$ with the completion of $A(G)$ at its augmentation ideal, a deep result in equivariant homotopy theory.