Equivariant Hurewicz Theorem

• The Equivariant Hurewicz Theorem extends the classical result by linking equivariant fundamental groupoids to first equivariant homology groups for spaces with group actions.
• It employs equivariant abelianisation and derived module techniques to convert homotopical information into computable homological invariants.
• The framework offers practical tools for classifying equivariant covering spaces and understanding how group symmetries influence topological structures.

An equivariant analog of the Hurewicz theorem generalizes the classical relationship between fundamental group(oid) and first homology to spaces with group actions, establishing a precise correspondence between equivariant homotopical and homological invariants. This formalism integrates equivariant fundamental groupoids, equivariant abelianisation, and derived modules, yielding isomorphism results under suitable conditions and providing foundational tools for the computation and understanding of equivariant invariants in algebraic topology and homotopy theory.

1. Classical Hurewicz Theorem and Its Groupoid-Relative Form

The classical (nonequivariant) Hurewicz theorem asserts that for a topological pair $(X,A)$ with $A$ totally path disconnected and intersecting every path component of $X$, there is an isomorphism between the total abelianisation of the fundamental groupoid and the relative first homology group: $\pi_1(X,A)^{\mathrm{totab}} \cong H_1(X,A)$ This is realized by identifying the canonical map $\pi_1(X,A) \to H_1(X,A)$ with the total abelianisation—namely, forming the quotient of the groupoid by the normal subgroupoid generated by all commutators, and relating this process to Crowell’s derived module construction, yielding exact sequences such as

$0 \to N \to D_\varphi \to IG \to 0$

where $G$ is the (possibly multi-basepoint) groupoid, $\varphi : F \to G$ is an epimorphism, $N = \ker \varphi$, and $IG$ is the augmentation module.

2. Equivariant Fundamental Groupoids

Given a space $X$ with a compact Lie group (or discrete group) $G$ acting and a $G$-invariant subspace $A$, one defines the equivariant fundamental groupoid $\pi_1^G(X, A)$ through several constructions:

• G-action on the groupoid: For $g \in G$, there is an induced automorphism $g : \pi_1(X, A) \to \pi_1(X, A)$.
• Groupoid on orbits: One may consider groupoids on the orbit space $X/G$, compatible with the original G-action.
• Action groupoid: Via the Grothendieck construction, the action groupoid $G \ltimes X$ encodes both the space and the group action.

Equivariant covering morphisms—maps which respect both the covering space structure and the $G$-action—generalize classical covering morphisms of groupoids and require unique path lifting, formulated respecting $G$-actions.

3. Equivariant Abelianisation

For a $G$-equivariant groupoid $\pi_1^G(X,A)$, the equivariant abelianisation is constructed by taking abelianisation in an equivariant manner—either by forming the quotient by commutators in a $G$-invariant way or via the orbit category $\mathcal{O}_G$. For the universal group $UG$ of a groupoid $G$,

$G^{\mathrm{ab}} \cong (UG)^{\mathrm{ab}}$

with the equivariant analog taking the $G$-fixed points or acting within the orbit category. The expected isomorphism for $G$-equivariant spaces, connecting equivariant homotopy to equivariant homology, is

$[\pi_1^G(X, A)]^{\mathrm{totab}} \cong H_1^G(X, A)$

where $H_1^G(X, A)$ denotes an equivariant homology theory, such as Borel or chain complex-based equivariant homology, incorporating the $G$-action.

4. Derived Modules in the Equivariant Setting

For an equivariant groupoid morphism $\phi : F \to G$ (with $F$ a $G$-module), one defines the equivariant derived module $D_\phi^G$, fitting into an exact sequence of $G$-modules: $0 \to N \to D_\phi^G \to I^G G \to 0$ where $I^G G$ denotes the equivariant augmentation module. This construction generalizes the non-equivariant theory and is crucial for classifying group extensions with operators, elucidating twists present in the transition from homotopy to homology in the equivariant context.

5. Isomorphism and the Equivariant Relative Hurewicz Theorem

The core equivariant analog of the dimension-1 relative Hurewicz theorem asserts that, under suitable conditions (notably, $A$ totally path disconnected and meeting every path component), the equivariant Hurewicz morphism: $\pi_1^G(X, A) \to H_1^G(X, A)$ induces an isomorphism

$\pi_1^G(X, A)^{\mathrm{totab}} \cong H_1^G(X, A)$

Here, the passage from the equivariant fundamental groupoid to homology is mediated by the equivariant abelianisation and the framework of derived modules. Covering morphisms, adapted to the $G$-action, allow the translation between equivariant "covering spaces" and the corresponding groupoids, maintaining the homotopical information within the category of $G$-spaces.

6. Explicit Formulations

Key formulas from this framework include:

• The exact sequence of derived modules in the equivariant case:

$0 \to N \to D_\phi^G \to I^G G \to 0$

• The universal abelianisation of a groupoid in terms of orbit representatives:

$$G^{\mathrm{totab}} \cong \bigoplus_{[i] \in \mathcal{O}_G} \left[G(a_{[i]}) \oplus F\right]$$

with $F$ a free abelian group from a tree subgroupoid and $[i]$ indexing $G$-orbits.

• The 1-dimensional equivariant Hurewicz isomorphism:

$\pi_1^G(X, A)^{\mathrm{totab}} \cong H_1^G(X, A)$

• The functorial nature of the equivariant Hurewicz morphism, which factors through equivariant derived modules in exact sequences analogous to the non-equivariant case.

7. Implications and Connections

This equivariant framework linearizes the equivariant fundamental groupoid, permitting the computation of equivariant first homology as the abelianization of first homotopy data—including group actions and symmetry. The impact extends to:

• Computation of equivariant homology groups
• Classification of equivariant covering spaces via groupoid approaches
• Systematic understanding of how symmetry (through group actions) interacts with topological invariants
• Relating group extensions with operators to equivariant homology via derived module formalism

These results—grounded in the methods of covering morphisms of groupoids, abelianisations, and derived modules—comprise a robust equivariant analog of the Relative Hurewicz Theorem in dimension 1, shaping the paper of symmetry in algebraic topology and homotopy theory (Brown, 2010).