[Paper Review] Global actions, groupoid atlases and related topics

Anthony Bak, Ronald Brown+2 more|ArXiv.org|Jun 11, 2006

Geometric and Algebraic Topology5 references19 citations

TL;DR

This paper introduces global actions and groupoid atlases as algebraic tools to model higher algebraic K-theory with combinatorial control over elementary operations, paths, and homotopies. It establishes that the Steinberg group $σ_n(R)$ is isomorphic to the universal simply connected covering of the global action $σ_n(R)$, proving $K_2(n,R) \cong \pi_1(\mathsf{GL}_n(R))$, thereby linking group cohomology and homotopy theory in a concrete algebraic setting.

ABSTRACT

A. Bak developed a combinatorial approach to higher $K$-theory, in which control is kept of the elementary operations involved, through paths and `paths of paths' in what he called a global action. The homotopy theory of these was developed by G. Minian. R. Brown and T. Porter developed applications to identities among relations for groups, and also the extension to groupoid atlases. This paper is intended as an introduction tothis circle of ideas, and so to give a basis for exploration and development of this area.

Motivation & Objective

• To develop a combinatorial framework for higher algebraic K-theory that preserves the geometric intuition of Whitehead’s work on simple homotopy theory.
• To generalize group actions into groupoid atlases, enabling a local-to-global structure analogous to manifolds but in algebraic settings.
• To provide a concrete, computable model of the fundamental group and covering spaces for algebraic groups like $\mathsf{GL}_n(R)$.
• To establish a precise isomorphism between the second $K$-group $K_2(n,R)$ and the fundamental group $\pi_1(\mathsf{GL}_n(R))$ via global action homotopy theory.

Proposed method

• Define a global action as a set with a family of group actions indexed by a poset, equipped with patching conditions to model local-to-global behavior.
• Construct the fundamental groupoid of a global action using curves, paths, and homotopies of paths, generalizing classical homotopy theory to algebraic settings.
• Introduce groupoid atlases by replacing group actions with action groupoids, allowing equivalence relations and resolutions to be modeled algebraically.
• Use nerve constructions and barycentric subdivision to relate global actions to simplicial complexes and compute homotopy invariants.
• Define the Steinberg global action $\mathsf{St}_n(R)$ as the colimit of local groups $\operatorname{St}_n(R)_\alpha$ over maximal chains in the poset $\Phi$, with relations restricted to $i<j$.
• Establish an isomorphism between $\mathsf{St}_n(R)$ and the universal covering of $\mathsf{GL}_n(R)$ via compatible isomorphisms on maximal local groups $\operatorname{GL}_n(R)_\alpha \cong T_n(R)$.

Experimental results

Research questions

• RQ1 How can global actions provide a combinatorial model of higher algebraic $K$-theory that retains the geometric intuition of Whitehead’s work?
• RQ2 What is the role of groupoid atlases in generalizing group actions and enabling the study of identities among relations in group presentations?
• RQ3 How can the fundamental group and covering spaces of a global action be constructed and computed in algebraic terms?
• RQ4 What is the precise relationship between the Steinberg group $\mathsf{St}_n(R)$ and the fundamental group of $\mathsf{GL}_n(R)$?
• RQ5 Can the second $K$-group $K_2(n,R)$ be realized as the fundamental group of a global action?

Key findings

• The Steinberg group $\mathsf{St}_n(R)$ is isomorphic to the universal simply connected covering of the global action $\mathsf{GL}_n(R)$, with the covering map given by the evaluation homomorphism $\varphi: \mathsf{St}_n(R) \to \mathsf{GL}_n(R)$.
• The kernel of the covering map $\varphi$ is central and trivial when restricted to any local group $\operatorname{GL}_n(R)_\alpha$, confirming the covering is simply connected.
• For maximal $\alpha$, the local group $\operatorname{GL}_n(R)_\alpha$ is isomorphic to the group $T_n(R)$ of upper triangular matrices, and this isomorphism extends compatibly across the poset $\Phi$.
• The colimit of the local groups $\operatorname{GL}_n(R)_\alpha$ over $\Phi$ is isomorphic to the group $\widetilde{\mathsf{GL}_n(R)}$, the universal cover of $\mathsf{E}_n(R)$.
• The fundamental group $\pi_1(\mathsf{GL}_n(R), 1)$ is isomorphic to the second $K$-group $K_2(n,R)$, establishing a direct homotopical interpretation of $K_2$.
• The global action $\mathsf{St}_n(R)$ is connected and its colimit over $\Phi$ is isomorphic to $\mathsf{St}_n(R)$, confirming its role as the universal cover.