G-Spaces and Equivariant Maps

Table of Contents

• #1. Basic Definitions|1. Basic Definitions
  • #1.1 The Category of G-Spaces via Monads|1.1 The Category of G-Spaces via Monads
  • #1.2 Explicit Definition and Morphisms|1.2 Explicit Definition and Morphisms
  • #1.3 Mapping Spaces and Enrichment|1.3 Mapping Spaces and Enrichment
  • #1.4 G-Homotopies|1.4 G-Homotopies
• #2. Fixed Points, Orbits, and Key Examples|2. Fixed Points, Orbits, and Key Examples
  • #2.1 Fixed-Point Spaces and Isotropy|2.1 Fixed-Point Spaces and Isotropy
  • #2.2 Corepresentability of Fixed Points|2.2 Corepresentability of Fixed Points
  • #2.3 Induction and Coinduction|2.3 Induction and Coinduction
  • #2.4 Representation Spheres|2.4 Representation Spheres
• #3. Naive vs. Genuine G-Spaces|3. Naive vs. Genuine G-Spaces
  • #3.1 Naive G-Spaces|3.1 Naive G-Spaces
  • #3.2 Genuine Weak Equivalences|3.2 Genuine Weak Equivalences
  • #3.3 The Model Structure on G Top|3.3 The Model Structure on G Top
  • #3.4 Equivariant Homotopy Groups|3.4 Equivariant Homotopy Groups
  • #3.5 The Unstable Distinction in Context|3.5 The Unstable Distinction in Context
• #4. G-CW Complexes|4. G-CW Complexes
  • #4.1 Cells and Skeleta|4.1 Cells and Skeleta
  • #4.2 Fixed Points of Cells|4.2 Fixed Points of Cells
  • #4.3 Colimit Compatibility|4.3 Colimit Compatibility
  • #4.4 Theta-Connectedness and the HELP Lemma|4.4 Theta-Connectedness and the HELP Lemma
  • #4.5 Worked Examples|4.5 Worked Examples
• #5. Elmendorf’s Theorem|5. Elmendorf’s Theorem
  • #5.1 The Orbit Category|5.1 The Orbit Category
  • #5.2 The Presheaf Functor|5.2 The Presheaf Functor
  • #5.3 Statement and Proof Sketch|5.3 Statement and Proof Sketch
  • #5.4 The Infinity-Categorical Statement|5.4 The Infinity-Categorical Statement
  • #5.5 Applications|5.5 Applications
• #References|References

1. Basic Definitions 📐

1.1 The Category of G-Spaces via Monads

Throughout, let \(G\) be a topological group — a group object in \(\mathbf{Top}\). We define the category \(G\mathbf{Top}\) of G-spaces as the category of algebras over a monad.

Definition (The Monad \(M_G\)). Define the endofunctor \(M_G: \mathbf{Top} \to \mathbf{Top}\) by

\[M_G(X) = G \times X.\]

The monad structure is given by: - Unit \(\eta_X: X \to G \times X\), \(x \mapsto (e, x)\) (inclusion at the identity), - Multiplication \(\mu_X: G \times G \times X \to G \times X\), \((g, h, x) \mapsto (gh, x)\).

Associativity and unitality of this monad follow directly from the group axioms of \(G\).

The category of \(M_G\)-algebras is exactly \(G\mathbf{Top}\): an algebra structure \(\alpha: G \times X \to X\) satisfying the algebra axioms is precisely a continuous \(G\)-action. The algebra maps are exactly equivariant maps.

For based G-spaces we instead work with the monad \(M_G^+\) on \(\mathbf{Top}_*\) sending \(X \mapsto G_+ \wedge X\), where \(G_+ = G \sqcup \{*\}\) is \(G\) with a disjoint basepoint. The based category is denoted \(G\mathbf{Top}_*\).

1.2 Explicit Definition and Morphisms

Unpacking the monad description gives the explicit definition.

Definition (G-Space). A G-space is a topological space \(X\) equipped with a continuous action map

\[\mu: G \times X \longrightarrow X, \quad (g, x) \mapsto g \cdot x,\]

satisfying: 1. (Associativity) \(g \cdot (h \cdot x) = (gh) \cdot x\) for all \(g, h \in G\), \(x \in X\). 2. (Unitality) \(e \cdot x = x\) for all \(x \in X\).

These conditions correspond precisely to \(\alpha \circ (m_G \times \mathrm{id}) = \alpha \circ (\mathrm{id}_G \times \alpha)\) and \(\alpha \circ (\eta \times \mathrm{id}) = \mathrm{id}_X\) in the monad algebra axioms.

Definition (Equivariant Map). A G-equivariant map (or G-map) between G-spaces \((X, \mu_X)\) and \((Y, \mu_Y)\) is a continuous map \(f: X \to Y\) such that

\[f(g \cdot x) = g \cdot f(x) \quad \text{for all } g \in G, x \in X,\]

i.e., the following diagram commutes:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
G \times X \arrow[r, "{\mathrm{id}_G \times f}"] \arrow[d, "\mu_X"'] & G \times Y \arrow[d, "\mu_Y"] \\
X \arrow[r, "f"'] & Y
\end{tikzcd}
\end{document}

\(G\mathbf{Top}\) denotes the category with objects G-spaces and morphisms G-equivariant maps.

[!TIP]- Hint for Exercise 1 A map \(f: X \to Y\) lies in the equalizer exactly when \(\mu_Y(g, f(x)) = f(\mu_X(g, x))\) for all \((g, x) \in G \times X\), which is the equivariance condition. Equalizers in \(\mathbf{Top}\) are closed inclusions of the underlying sets with the subspace topology.

1.3 Mapping Spaces and Enrichment

Three distinct notions of mapping space appear naturally in \(G\mathbf{Top}\), and keeping them distinct is essential.

Definition (Fixed-Point Mapping Space). For G-spaces \(X\) and \(Y\), the equivariant mapping space is

\[\mathrm{Map}_G(X, Y) = \{ f \in \mathrm{Map}(X, Y) \mid f \text{ is } G\text{-equivariant} \}\]

with the subspace topology inherited from \(\mathrm{Map}(X, Y)\) (compactly generated compact-open topology). This is the internal hom of the fixed-point enrichment.

Definition (Conjugation Action). The conjugation G-space of maps is

\[G\mathrm{Map}(X, Y) = \mathrm{Map}(X, Y)\]

as a topological space, equipped with the conjugation action:

\[(g \cdot f)(x) = g \cdot f(g^{-1} \cdot x).\]

The self-enrichment of \(G\mathbf{Top}\) uses \(G\mathrm{Map}(X, Y)\) as the internal hom object — the category \(G\mathbf{Top}\) is enriched over itself.

Key Relation. The equivariant mapping space is the fixed points of the conjugation space:

\[\mathrm{Map}_G(X, Y) = \bigl(G\mathrm{Map}(X, Y)\bigr)^G.\]

Proof. A map \(f \in G\mathrm{Map}(X,Y)\) is fixed by every \(g \in G\) under conjugation iff \(g \cdot f = f\) for all \(g\), i.e., \(g \cdot f(g^{-1} x) = f(x)\) for all \(g, x\). Setting \(x' = g^{-1} x\) gives \(f(gx') = g \cdot f(x')\), which is exactly the equivariance condition. \(\square\)

[!TIP]- Hint for Exercise 2 For (a), compute \((gh) \cdot f\) explicitly and compare with \(g \cdot (h \cdot f)\) — both sides simplify to \(x \mapsto gh \cdot f(h^{-1}g^{-1}x)\). For (b), write out the fixed-point condition \(g \cdot f = f\) pointwise and substitute \(x' = g^{-1}x\).

1.4 G-Homotopies

Definition (G-Homotopy). A G-homotopy between G-maps \(f_0, f_1: X \to Y\) is a continuous map

\[h: X \times I \longrightarrow Y\]

that is a morphism in \(G\mathbf{Top}\), where \(G\) acts on \(X \times I\) via \(g \cdot (x, t) = (g \cdot x, t)\) — that is, \(G\) acts trivially on \(I = [0,1]\).

The triviality of the \(G\)-action on \(I\) is forced: we want homotopies to be “the same at every time \(t\)” from \(G\)’s perspective. Explicitly, \(h\) is a G-homotopy iff \(h(g \cdot x, t) = g \cdot h(x, t)\) for all \(g \in G\), \(x \in X\), \(t \in I\).

2. Fixed Points, Orbits, and Key Examples 🔑

2.1 Fixed-Point Spaces and Isotropy

Definition (Fixed-Point Space). For a G-space \(X\) and a subgroup \(H \leq G\), the \(H\)-fixed-point subspace is

\[X^H = \{ x \in X \mid h \cdot x = x \text{ for all } h \in H \}.\]

This is a closed subspace of \(X\) (since it is the equalizer of the continuous maps \(\mu(h, -): X \to X\) and \(\mathrm{id}_X\) for each \(h \in H\), and a countable intersection of closed sets when \(H\) is second countable).

Definition (Weyl Group). The Weyl group of \(H\) in \(G\) is

\[WH = NH/H,\]

where \(NH = \{ g \in G \mid gHg^{-1} = H \}\) is the normalizer of \(H\). The Weyl group \(WH\) acts on \(X^H\) by \([n] \cdot x = n \cdot x\), making \(X^H\) naturally a \(WH\)-space.

[!TIP]- Hint for Exercise 3 Since \(H \unlhd G\), we have \(NH = G\), so \(WH = G/H\). The proposed \(G/H\)-action is \([g] \cdot x := g \cdot x\); check independence of representative: if \(g' = gh\) for \(h \in H\), then \(g' \cdot x = gh \cdot x = g \cdot (h \cdot x) = g \cdot x\) since \(x \in X^H\).

Definition (Isotropy Group). For \(x \in X\), the isotropy group (or stabilizer) of \(x\) is

\[G_x = \{ g \in G \mid g \cdot x = x \} \leq G.\]

The orbit of \(x\) is \(Gx = G/G_x\) as a \(G\)-set, and the orbit map \(G \to Gx\), \(g \mapsto gx\), induces a homeomorphism \(G/G_x \xrightarrow{\sim} Gx\) when \(G\) is compact and \(X\) is Hausdorff.

[!TIP]- Solution to Exercise 4 The subgroups of \(C_4\) are \(\{e\} \subset C_2 = \langle r^2 \rangle \subset C_4\). Fixed-point sets: - \(X^{\{e\}} = S^1\) (nothing is fixed by the trivial subgroup) - \(X^{C_2}\): need \(r^2 \cdot z = -z = z\), i.e., \(z = 0\) — but \(0 \notin S^1\). So \(X^{C_2} = \emptyset\). - \(X^{C_4} \subseteq X^{C_2} = \emptyset\), so \(X^{C_4} = \emptyset\).

Isotropy: a generic \(z \in S^1\) (not a 4th root of unity in a special sense) has \(G_z = \{e\}\). The orbit of any \(z\) has 4 elements \(\{z, iz, -z, -iz\}\), so \(|G/G_z| = 4\), confirming \(G_z = \{e\}\) for all \(z \in S^1\).

2.2 Corepresentability of Fixed Points

The orbit spaces \(G/H\) play a fundamental role: they corepresent the fixed-point functors.

Proposition (Corepresentability). For any G-space \(X\) and closed subgroup \(H \leq G\),

\[X^H \cong \mathrm{Map}_G(G/H, X).\]

Proof. A \(G\)-equivariant map \(f: G/H \to X\) is determined by \(f(eH) \in X\), subject to the condition that \(f(eH)\) is fixed by \(H\): for \(h \in H\), equivariance gives \(h \cdot f(eH) = f(h \cdot eH) = f(eH)\). Conversely, any \(x \in X^H\) determines an equivariant map \(f_x: G/H \to X\) by \(f_x(gH) = g \cdot x\) (well-defined since \(x \in X^H\)). These assignments are inverse homeomorphisms. \(\square\)

[!TIP]- Hint for Exercise 5 Apply corepresentability with \(X = G/K\): a \(G\)-map \(G/H \to G/K\) is determined by the image of the coset \(eH\), which must land in \((G/K)^H\). A coset \(gK \in G/K\) is fixed by \(h \in H\) iff \(hgK = gK\) iff \(g^{-1}hg \in K\).

2.3 Induction and Coinduction

Let \(H \leq G\) be a closed subgroup. The forgetful functor \(U: G\mathbf{Top} \to H\mathbf{Top}\) restricts the group action from \(G\) to \(H\). This functor has both a left and a right adjoint.

Definition (Induced G-Space). The induced G-space of an \(H\)-space \(X\) is the balanced product

\[G \times_H X = (G \times X) / {\sim}, \quad (gh, x) \sim (g, hx) \text{ for } h \in H,\]

with \(G\)-action \(g' \cdot [g, x] = [g'g, x]\). This is the left adjoint of \(U\):

\[G\mathbf{Top}(G \times_H X, Y) \cong H\mathbf{Top}(X, UY).\]

Definition (Coinduced G-Space). The coinduced G-space of an \(H\)-space \(X\) is

\[\mathrm{Map}_H(G, X),\]

the space of \(H\)-equivariant maps \(G \to X\) (where \(H\) acts on \(G\) by left multiplication), with \(G\)-action \((g \cdot f)(g') = f(g'g)\) (right translation). This is the right adjoint of \(U\):

\[H\mathbf{Top}(UY, X) \cong G\mathbf{Top}(Y, \mathrm{Map}_H(G, X)).\]

[!TIP]- Hint for Exercise 6 Given \(f: G \times_H X \to Y\), set \(\tilde{f}(x) = f([e, x])\). For the converse, set \(\hat{g}([g', x]) = g' \cdot g(x)\) and check this is independent of the representative: if \((g'h, x) \sim (g', hx)\), then \(g'h \cdot g(x) = g' \cdot (h \cdot g(x)) = g' \cdot g(hx)\) using \(H\)-equivariance of \(g\).

[!EXAMPLE]- Induction for \(C_2 \leq C_4\) Let \(G = C_4 = \langle r \mid r^4 = 1 \rangle\) and \(H = C_2 = \langle r^2 \rangle\). Let \(X = \{pt\}\) with trivial \(H\)-action.

Then \(G \times_H X = C_4 \times_{C_2} \{pt\} \cong C_4/C_2 \cong \{eC_2, rC_2\}\), which is a two-point \(C_4\)-space — the orbit \(C_4/C_2\).

On the other hand, \(\mathrm{Map}_H(G, X) = \mathrm{Map}_{C_2}(C_4, \{pt\}) = \{pt\}\) (only one map).

[!TIP]- Solution to Exercise 7 (a) As a set, \(C_4 \times_{C_2} S^1 = (C_4 \times S^1)/{\sim}\) where \((r^2 g, z) \sim (g, -z)\). Since \(C_4/C_2 = \{[e], [r]\}\), we can write every element as \([e, z]\) or \([r, z]\) with \(z \in S^1\). The identification is \([e, -z] = [r^2 \cdot e, z] \sim [e, r^2 \cdot z]\), but since \(r^2\) acts by \(-1\): \([e, -z]\) and \([e, z]\) are different elements, so the space is \(S^1 \sqcup S^1\) (two disjoint copies).

The \(C_4\)-action swaps the two copies: \(r \cdot [e, z] = [r, z]\) and \(r \cdot [r, z] = [r^2, z] = [e, -z]\). So \(C_4\) acts on \(S^1 \sqcup S^1\) by: \(r\) maps the first copy to the second and the second back to the first (with a \(-1\) twist).

2. \((C_4 \times_{C_2} S^1)^{C_4}\): a fixed point \([g, z]\) satisfies \(r \cdot [g, z] = [g, z]\). From the action above, no point is fixed, so \((C_4 \times_{C_2} S^1)^{C_4} = \emptyset\).

2.4 Representation Spheres

A key family of examples arises from linear representations.

Definition (Representation Sphere). Let \(V\) be a finite-dimensional real \(G\)-representation (i.e., a finite-dimensional real vector space with a continuous linear \(G\)-action). The representation sphere is

\[S^V = V^+ = V \cup \{\infty\},\]

the one-point compactification of \(V\), with the \(G\)-action extended so that \(G\) fixes \(\infty\).

Representation spheres are fundamental for defining \(RO(G)\)-graded cohomology theories, where one suspends not just by \(S^n\) but by arbitrary representation spheres \(S^V\).

[!TIP]- Solution to Exercise 8 \(S^{n\sigma}\) is the one-point compactification of \(\mathbb{R}^n\) with \(C_2\) acting by \(v \mapsto -v\). The fixed set \((S^{n\sigma})^{C_2}\) consists of points fixed by \(\tau\): \(v = -v\) iff \(v = 0\), plus the fixed point at \(\infty\) (since \(\tau\) fixes \(\infty\)). So \((S^{n\sigma})^{C_2} = \{0, \infty\} \cong S^0\) for all \(n \geq 1\), and \((S^0)^{C_2} = S^0\) (both points \(\pm 1\) are fixed since the \(C_2\)-action on \(S^0 = \{1,-1\}\) is trivial for \(0\)-dimensional representation).

For the smash product: \(S^\sigma \wedge S^\sigma \cong S^{2\sigma}\). Fixed sets: \((S^\sigma)^{C_2} = S^0\) and \((S^{2\sigma})^{C_2} = S^0\), so both sides are \(S^0\) — consistent with \(S^0 \wedge S^0 \cong S^0\).

3. Naive vs. Genuine G-Spaces 🎯

This section contains the most conceptually important distinction in equivariant homotopy theory.

3.1 Naive G-Spaces

Definition (Naive G-Space). A naive G-space is simply a topological space with a continuous \(G\)-action, viewed through the lens of \(\mathbf{Top}^{BG}\) — i.e., as a functor \(BG \to \mathbf{Top}\).

In the naive theory, a map \(f: X \to Y\) of G-spaces is a weak equivalence if the underlying map of topological spaces \(Uf: UX \to UY\) is a weak homotopy equivalence — that is, \(\pi_n(f): \pi_n(X) \xrightarrow{\sim} \pi_n(Y)\) for all \(n \geq 0\) and all basepoints. The \(G\)-action is completely ignored.

The naive theory is appropriate when one only cares about spaces parametrized by the classifying space \(BG\), not about the equivariant structure itself.

3.2 Genuine Weak Equivalences

Definition (Genuine Weak Equivalence). A map \(f: X \to Y\) in \(G\mathbf{Top}\) is a genuine weak equivalence if for every closed subgroup \(H \leq G\), the induced map on \(H\)-fixed points

\[f^H: X^H \longrightarrow Y^H\]

is a weak homotopy equivalence.

The key principle: genuine equivariant homotopy theory sees a G-space \(X\) as a system of spaces \(\{X^H\}_{H \leq G}\) varying over the lattice of closed subgroups, and requires weak equivalences to be detected at every level of this system simultaneously.

The distinction between naive and genuine weak equivalences is the central divide in equivariant homotopy theory:

flowchart TD
    A["G-map f: X → Y"]
    B["Underlying map Uf: UX → UY<br/>is a weak equivalence"]
    C["Fixed-point maps f^H: X^H → Y^H<br/>are weak equivalences for ALL H"]
    D["Naive weak equivalence"]
    E["Genuine weak equivalence"]
    A --> B --> D
    A --> C --> E
    E -->|"implies"| D
    D -->|"does NOT imply"| E

The implication is strict. Every genuine weak equivalence is a naive weak equivalence (taking \(H = \{e\}\) gives the underlying map), but the converse fails, as the \(C_2\)-action example above demonstrates.

[!TIP]- Solution to Exercise 9 (a) The underlying spaces are both \(S^1\), so the identity is a homeomorphism, hence a weak equivalence naively.

2. \(X^{C_2} = \{e^{i\theta} \mid e^{-i\theta} = e^{i\theta}\} = \{1, -1\} \cong S^0\) (the two points where \(\sin\theta = 0\)).

\(Y^{C_2} = \{e^{i\theta} \mid -e^{i\theta} = e^{i\theta}\} = \emptyset\) (no point satisfies \(z = -z\) on \(S^1 \subset \mathbb{C} \setminus \{0\}\)).

Since \(X^{C_2} \simeq S^0 \not\simeq \emptyset = Y^{C_2}\), the identity is not a genuine weak equivalence. The two \(C_2\)-spaces have completely different equivariant homotopy types.

3.3 The Model Structure on G Top

The genuine weak equivalences are the weak equivalences in a model structure on \(G\mathbf{Top}\).

Proposition 1.2.15 (Genuine Model Structure on \(G\mathbf{Top}\)). There is a cofibrantly generated model structure on \(G\mathbf{Top}\) in which:

• Weak equivalences are genuine weak equivalences: maps \(f: X \to Y\) with \(f^H: X^H \xrightarrow{\sim} Y^H\) for all closed \(H \leq G\).
• Fibrations are maps \(p: X \to Y\) such that \(p^H: X^H \to Y^H\) is a Serre fibration for all closed \(H \leq G\).
• Cofibrations are retracts of relative \(G\)-CW complexes (see §4).

The generating cofibrations are:

\[I_G = \{ G/H \times S^{n-1} \hookrightarrow G/H \times D^n \mid H \leq G \text{ closed}, n \geq 0 \}.\]

The generating acyclic cofibrations are:

\[J_G = \{ G/H \times D^n \hookrightarrow G/H \times D^n \times I \mid H \leq G \text{ closed}, n \geq 0 \}.\]

3.4 Equivariant Homotopy Groups

Definition (Equivariant Homotopy Groups). For a based G-space \(X\) with \(H\)-fixed basepoint, the \(H\)-equivariant homotopy groups are

\[\pi_n^H(X) := \pi_n(X^H),\]

the ordinary homotopy groups of the \(H\)-fixed-point space.

These are indexed not just by \(n \geq 0\) but by the lattice of closed subgroups of \(G\):

\[\bigl\{ \pi_n^H(X) \bigr\}_{H \leq G,\, n \geq 0}.\]

The genuine weak equivalences are precisely the maps inducing isomorphisms on all \(\pi_n^H\). This is the equivariant analog of Whitehead’s theorem characterizing weak equivalences via homotopy groups.

3.5 The Unstable Distinction in Context 💡

The naive/genuine distinction is worth pausing on, because it is far less dramatic in the unstable setting than in the stable one — and this asymmetry explains why it receives comparatively little attention in unstable equivariant homotopy theory.

Unstable: same objects, different weak equivalences. Both the naive and genuine theories live on the same category \(G\mathbf{Top}\) — spaces with continuous \(G\)-action. They are two different model structures on the same underlying category:

The naive homotopy category is equivalent to the homotopy theory of spaces parametrized over \(BG\) — it sees only the total homotopy type and the monodromy action of \(\pi_1(BG) = G\) on it. All finer fixed-point information is lost.

Why genuine is the right answer unstably. Elmendorf’s theorem (§5) makes the genuine theory completely natural: genuine G-spaces are presheaves of spaces on \(\mathcal{O}_G\), and the weak equivalences are just objectwise equivalences. There is no comparably clean description of the naive homotopy theory in terms of the equivariant structure. The naive theory is a coarsening of the genuine one, obtained by restricting to the single object \(G/e \in \mathcal{O}_G\) (the free orbit).

Contrast with the stable world. In the stable world, naive and genuine \(G\)-spectra are different categories, not just different model structures:

• Naive \(G\)-spectra: ordinary spectra with a \(G\)-action (i.e., objects of \(\mathrm{Sp}^{BG}\), spectra parametrized over \(BG\)). Homotopy groups \(\pi_n^H(X) = \pi_n(X^{hH})\) (homotopy fixed points).
• Genuine \(G\)-spectra: spectra indexed over a complete \(G\)-universe \(\mathcal{U}\) (a countably infinite-dimensional real \(G\)-representation containing all irreducibles). Homotopy groups are \(RO(G)\)-graded: \(\pi_V^H(X) = [S^V, X]^H\) for \(V \in RO(G)\).

The difference is structural: genuine \(G\)-spectra have norm maps \(N_H^G: \mathrm{Sp}^H \to \mathrm{Sp}^G\) — multiplicative transfers that have no naive analogue. These norms are essential in the Hill–Hopkins–Ravenel theorem resolving the Kervaire invariant one problem (2009): the key object is the genuine \(C_8\)-spectrum \(\Omega = D^{-1} MU^{((C_8))}\), and the proof relies critically on norm maps and \(RO(G)\)-graded homotopy groups that are unavailable in the naive stable theory.

4. G-CW Complexes 🏗️

4.1 Cells and Skeleta

The cells in a G-CW complex are orbit spaces times discs.

Definition (G-Cells). The \(n\)-dimensional \(G\)-cells are spaces of the form

\[G/H \times D^n \quad \text{(interior cell)}, \qquad G/H \times S^{n-1} \quad \text{(boundary cell)},\]

where \(H \leq G\) is a closed subgroup and \(G\) acts on the orbit factor \(G/H\) by left translation and trivially on \(D^n\) and \(S^{n-1}\).

Definition 1.2.1 (G-CW Complex). A G-CW complex is a G-space \(X\) equipped with a filtration

\[X^0 \subseteq X^1 \subseteq X^2 \subseteq \cdots \subseteq X = \operatorname{colim}_n X^n\]

where: - \(X^0 = \coprod_\alpha G/H_\alpha\) is a disjoint union of orbits (a discrete set of \(G\)-orbits), - \(X^{n+1}\) is obtained from \(X^n\) by a pushout of the form:

\[\begin{array}{ccc} \displaystyle\coprod_\beta G/H_\beta \times S^n & \hookrightarrow & \displaystyle\coprod_\beta G/H_\beta \times D^{n+1} \\[4pt] \downarrow & & \downarrow \\ X^n & \longrightarrow & X^{n+1} \end{array}\]

where the \(H_\beta\) range over (possibly varying) closed subgroups of \(G\).

The horizontal top map is the standard inclusion \(S^n \hookrightarrow D^{n+1}\), and the left vertical map is the attaching map of the cells.

[!TIP]- Solution to Exercise 10 Take \(n\) equally spaced points \(\zeta^k\) for \(k = 0, \ldots, n-1\) as the \(0\)-skeleton: these form a single free orbit \(C_n/e\) (since no point is fixed by any nontrivial rotation). Then attach a single \(1\)-cell of type \(C_n/e \times D^1\): the \(n\) arcs between consecutive points, all in the same free orbit. The resulting \(C_n\)-CW structure has one \(0\)-cell of type \(C_n/e\) and one \(1\)-cell of type \(C_n/e \times D^1\).

Fixed points: \((S^1)^{C_n}\) requires \(e^{2\pi i/n} z = z\), i.e., \(z = 0\), but \(0 \notin S^1\). So \((S^1)^{C_n} = \emptyset\), consistent with all cells being free.

[!TIP]- Solution to Exercise 11 Since the antipodal map has no fixed points, every cell must be free (orbit type \(C_2/e\)). One minimal structure: - Two \(0\)-cells forming one free orbit: \(C_2/e \times D^0\) (pick the north pole \(N\) and south pole \(S = \tau(N)\), but the antipodal of \(N = (0,0,1)\) is \(S = (0,0,-1)\) — these form a free orbit). - One free \(1\)-cell \(C_2/e \times D^1\): a great semicircle from \(N\) to \(S\) and its antipodal image, forming a free orbit of arcs. - One free \(2\)-cell \(C_2/e \times D^2\): the two hemispheres bounding the above.

Compare: the beachball structure for \(C_2\) acting by rotation (not antipodal) has two fixed \(0\)-cells of type \(C_2/C_2\); here all cells are free.

[!EXAMPLE]- Zero-Dimensional G-CW Complexes A \(0\)-dimensional \(G\)-CW complex is simply a disjoint union of orbits: \[X^0 = \coprod_\alpha G/H_\alpha.\] These are the “discrete G-spaces” — their fixed-point sets are \((G/H_\alpha)^K = \mathrm{Map}_G(G/K, G/H_\alpha)\), which is nonempty iff some \(G\)-conjugate of \(K\) is contained in \(H_\alpha\).

4.2 Fixed Points of Cells

To understand the homotopy theory, one must understand the fixed-point sets of cells.

Lemma (Fixed Points of a Cell). For closed subgroups \(H, K \leq G\),

\[(G/K \times D^n)^H = (G/K)^H \times D^n.\]

Proof. The \(G\)-action is \((G/K \times D^n)\) with trivial action on \(D^n\), so \(H\) fixes a pair \((gK, x)\) iff \(H\) fixes \(gK\) (and \(x\) is arbitrary). \(\square\)

This reduces the computation of cell fixed points to understanding \((G/K)^H\).

Proposition. There is a natural bijection

\[(G/K)^H = \mathrm{Map}_G(G/H, G/K).\]

Proof. By the corepresentability result of §2.2, \(\mathrm{Map}_G(G/H, G/K) \cong (G/K)^H\). \(\square\)

[!TIP]- Solution to Exercise 12 \((G/K)^H = \{gK \mid g^{-1}Hg \subseteq K\}\). Since \(C_4\) is abelian, \(g^{-1}Hg = H\) for all \(g\), so the condition is simply \(H \subseteq K\).

	\(K = \{e\}\)	\(K = C_2\)	\(K = C_4\)
\(H = \{e\}\)	\(C_4/\{e\} \cong C_4\) (4 elements)	\(C_4/C_2 \cong \{eC_2, rC_2\}\) (2 elements)	\(C_4/C_4 = \{*\}\) (1 element)
\(H = C_2\)	\(\emptyset\) (\(C_2 \not\subseteq \{e\}\))	\(C_4/C_2\) (2 elements, since \(C_2 \subseteq C_2\))	\(\{*\}\)
\(H = C_4\)	\(\emptyset\)	\(\emptyset\)	\(\{*\}\)

The set is nonempty exactly when \(H \subseteq K\).

4.3 Colimit Compatibility

A crucial technical fact underlies the model structure verification.

Proposition 1.2.8 (Fixed Points Commute with Relevant Colimits). Let \(H \leq G\) be a closed subgroup. The fixed-point functor \((-)^H: G\mathbf{Top} \to \mathbf{Top}\) commutes with: 1. Pushouts along closed inclusions (i.e., if \(A \hookrightarrow X\) is a \(G\)-equivariant closed inclusion and \(A \hookrightarrow B\) is any \(G\)-map, then \((X \cup_A B)^H \cong X^H \cup_{A^H} B^H\)). 2. Sequential colimits along closed inclusions (i.e., if \(X_0 \hookrightarrow X_1 \hookrightarrow \cdots\) are \(G\)-equivariant closed inclusions, then \((\operatorname{colim}_n X_n)^H \cong \operatorname{colim}_n (X_n^H)\)).

Corollary. If \(X\) is a \(G\)-CW complex, then \(X^H\) is a CW complex with cells \((G/K)^H \times D^n\) for each \(G\)-cell \(G/K \times D^n\) of \(X\). The CW filtration on \(X^H\) is induced from the skeletal filtration of \(X\).

4.4 Theta-Connectedness and the HELP Lemma

Definition 1.2.10 (\(\theta\)-Connected Maps). Let \(\theta: \{\text{closed subgroups of } G\} \to \mathbb{Z}_{\geq 0} \cup \{\infty\}\) be a function. A \(G\)-map \(f: X \to Y\) is \(\theta\)-connected if \(f^H: X^H \to Y^H\) is \(\theta(H)\)-connected for all closed subgroups \(H \leq G\).

Theorem 1.2.11 (Equivariant HELP Lemma — Homotopy Extension Lifting Property). Let \(f: X \to Y\) be a \(\theta\)-connected \(G\)-map and let \(i: A \hookrightarrow B\) be an inclusion of \(G\)-CW complexes such that all cells of \(B \setminus A\) of type \(G/H\) have dimension \(\leq \theta(H)\). Then \(f\) has the homotopy lifting property with respect to \(i\).

Proof sketch. By induction over the skeletal filtration of \(B\), using Proposition 1.2.8 to reduce each attachment to an ordinary (non-equivariant) HELP lemma problem for the fixed-point maps \(f^H\). \(\square\)

Corollary 1.2.14 (Equivariant Whitehead Theorem). A genuine weak equivalence between \(G\)-CW complexes is a \(G\)-homotopy equivalence.

Proof sketch. Apply the Equivariant HELP Lemma with \(A = \emptyset\), \(B = Y\), and \(f: X \to Y\) the weak equivalence. The \(\theta\)-connectedness condition is satisfied for \(\theta \equiv \infty\). Construct the inverse homotopy equivalence and homotopies inductively. \(\square\)

4.5 Worked Examples

[!EXAMPLE]- The Beachball: \(C_2\) Acting on \(S^V\) Let \(V = \mathbb{R}^2\) with \(C_2 = \{1, \tau\}\) acting by \(\tau \cdot (x,y) = (-x, -y)\) (the antipodal map). Then \(S^V = S^2\).

The \(C_2\)-CW structure on \(S^2\) (the “beachball”) is: - Two \(0\)-cells: \(\{N, S\}\) — one \(C_2\)-orbit \(C_2/e\), or equivalently two fixed points if \(\tau(N) = S\). Let \(\tau\) act antipodally, so \(\{N,S\}\) is the orbit \(C_2/e\). - One equatorial \(1\)-cell: \(C_2/e \times D^1\) — the equator as an orbit of arcs. - One \(2\)-cell: \(C_2/e \times D^2\) — the two hemispheres as an orbit.

The fixed-point set \((S^V)^{C_2}\) is the set of antipodally-fixed points of \(S^2\): since antipodal map has no fixed points on \(S^2\), \((S^V)^{C_2} = \emptyset\).

Compare with \(S^{2\sigma}\) where \(\sigma\) is the sign rep on \(\mathbb{R}\): here \(S^{2\sigma} = S^2\) with \(C_2\) reflecting across the equator. Then \((S^{2\sigma})^{C_2} = S^1\) (the equator).

[!EXAMPLE]- The Torus with \(S^1\)-Action Consider \(T^2 = S^1 \times S^1\) with the \(S^1\)-action \(z \cdot (w_1, w_2) = (zw_1, w_2)\) (rotation on the first factor).

A minimal \(S^1\)-CW structure requires only: - \(0\)-cells: One orbit \(S^1/S^1 \cong \{*\}\) — the point \(\{1\} \times S^1\) (fixed-point set), thought of as a single \(S^1\)-fixed point. Actually the fixed set is \((T^2)^{S^1} = \{1\} \times S^1\).

Wait — we need cells to build the whole torus. A minimal structure is: - One \(0\)-cell of type \(S^1/e\) (a free orbit) - One \(1\)-cell of type \(S^1/e \times D^1\)

This gives \(S^1 \cup_{S^1} S^1 \times I \cong T^2\). The remarkable fact is that the torus only needs a single free \(1\)-cell in its \(S^1\)-CW structure, whereas without the group action one needs \(2\)-cells as well.

[!EXAMPLE]- Equilateral Triangle with \(D_6\)-Action The dihedral group \(D_6 = \langle r, s \mid r^3 = s^2 = 1, srs = r^{-1} \rangle\) acts on the equilateral triangle \(T\) (as a topological space, homeomorphic to \(S^1\)).

The \(D_6\)-CW structure: - \(0\)-cells: The three vertices form a single \(D_6\)-orbit \(D_6/D_2\) (where \(D_2\) is the stabilizer of a vertex, isomorphic to \(\mathbb{Z}/2\), generated by the reflection fixing that vertex). - \(1\)-cells: The three edges form a single \(D_6\)-orbit \(D_6/\mathbb{Z}/2\) (where \(\mathbb{Z}/2\) is the reflection fixing the midpoint of an edge). But an edge has a midpoint fixed by a reflection, so the attaching data is an equivariant attachment.

The key point is that \(|D_6|/|D_2| = 6/2 = 3\) (the number of vertices), confirming the orbit count.

5. Elmendorf’s Theorem 🌐

5.1 The Orbit Category

Definition (Orbit Category). The orbit category \(\mathcal{O}_G\) is the full subcategory of \(G\mathbf{Top}\) on objects \(\{G/H \mid H \leq G \text{ closed}\}\).

The morphism spaces in \(\mathcal{O}_G\) are computed as:

\[\mathrm{Map}_{\mathcal{O}_G}(G/H, G/K) = \mathrm{Map}_G(G/H, G/K) \cong (G/K)^H.\]

By the analysis in §4.2, an element of \((G/K)^H\) is a coset \(gK\) with \(g^{-1}Hg \subseteq K\) — i.e., an element exhibiting a subconjugacy relation \(H \lesssim K\).

For \(G\) a finite group, \(\mathcal{O}_G\) is a finite category and this becomes an explicit finite combinatorial object.

[!TIP]- Solution to Exercise 13 Objects: \(C_p/e\) and \(C_p/C_p = \{*\}\). Morphism sets (using \((G/K)^H\)): - \(\mathrm{Hom}(C_p/e, C_p/e) = (C_p/e)^e = C_p/e \cong C_p\) — exactly \(p\) endomorphisms (the \(p\) elements of \(C_p\) acting by left multiplication). - \(\mathrm{Hom}(C_p/e, C_p/C_p) = (C_p/C_p)^e = \{*\}\) — one map (collapsing to the point). - \(\mathrm{Hom}(C_p/C_p, C_p/e) = (C_p/e)^{C_p} = \emptyset\) — no maps (no \(C_p\)-fixed points in \(C_p/e\)). - \(\mathrm{Hom}(C_p/C_p, C_p/C_p) = \{*\}\) — the identity.

So \(\mathcal{O}_{C_p}\) has two objects, a \(C_p\)-torsor of endomorphisms on the free orbit, one map to the fixed-point object, and no maps back. This is the archetype of all orbit categories.

[!TIP]- Solution to Exercise 14 From Exercise 12 (since \(C_4\) is abelian, \((G/K)^H = G/K\) if \(H \subseteq K\) and \(\emptyset\) otherwise):

	\(G/\{e\}\)	\(G/C_2\)	\(G/C_4\)
from \(G/\{e\}\)	\(C_4\) (4 maps)	\(C_4/C_2 \cong \mathbb{Z}/2\) (2 maps)	\(\{*\}\) (1 map)
from \(G/C_2\)	\(\emptyset\)	\(C_4/C_2\) (2 maps)	\(\{*\}\) (1 map)
from \(G/C_4\)	\(\emptyset\)	\(\emptyset\)	\(\{*\}\) (1 map)

Composition: the unique map \(G/\{e\} \to G/C_4\) factors through both \(G/C_2\) routes; the 4 endomorphisms of \(G/\{e\}\) compose as the group \(C_4\).

5.2 The Presheaf Functor

Every G-space \(X\) determines a presheaf on \(\mathcal{O}_G\).

Definition (Fixed-Point System Functor). Define the functor

\[\psi: G\mathbf{Top} \longrightarrow \mathrm{Fun}(\mathcal{O}_G^{\mathrm{op}}, \mathbf{Top})\]

by

\[\psi(X) = \bigl( G/H \mapsto X^H \bigr).\]

The functoriality in \(\mathcal{O}_G^{\mathrm{op}}\) is given as follows: a morphism \(\phi: G/H \to G/K\) in \(\mathcal{O}_G\) (corresponding to \(gK \in (G/K)^H\)) induces a restriction map

\[\phi^*: X^K \longrightarrow X^H, \quad x \mapsto g \cdot x.\]

(This is well-defined: if \(x \in X^K\) and \(h \in H\), then \(h \cdot (gx) = g \cdot (g^{-1}hg) \cdot x = g \cdot x\) since \(g^{-1}hg \in K\).)

Key Observation. Under \(\psi\), genuine weak equivalences in \(G\mathbf{Top}\) correspond exactly to objectwise weak equivalences in \(\mathrm{Fun}(\mathcal{O}_G^{\mathrm{op}}, \mathbf{Top})\) (the projective model structure): a map \(f: X \to Y\) is a genuine weak equivalence iff \(\psi(f)(G/H): X^H \to Y^H\) is a weak equivalence for all \(H\). This is the precise sense in which genuine G-spaces are “systems of spaces.”

[!TIP]- Solution to Exercise 15 (a) \(\psi(X)(C_2/e) = X^e = S^1\) (the whole space). \(\psi(X)(C_2/C_2) = X^{C_2}\) = fixed points of \(z \mapsto \bar{z}\) on \(S^1\): these are the real points \(\{z \in S^1 \mid \bar{z} = z\} = \{1, -1\} \cong S^0\).

2. The unique morphism \(C_2/e \to C_2/C_2\) (collapsing \(C_2\) to a point) induces the restriction \(X^{C_2} \hookrightarrow X^e\), i.e., the inclusion \(S^0 = \{1, -1\} \hookrightarrow S^1\).

So the presheaf \(\psi(X)\) is: the pair of spaces \((S^1, S^0)\) with the inclusion \(S^0 \hookrightarrow S^1\) as the structure map. This is all the homotopy-theoretic data of the \(C_2\)-space \(S^1\) with reflection.

5.3 Statement and Proof Sketch

Theorem 1.3.6 (Elmendorf, 1983). The functor

\[\psi: G\mathbf{Top} \longrightarrow \mathrm{Fun}(\mathcal{O}_G^{\mathrm{op}}, \mathbf{Top})\]

is the right adjoint of a Quillen equivalence. The left adjoint \(\Phi: \mathrm{Fun}(\mathcal{O}_G^{\mathrm{op}}, \mathbf{Top}) \to G\mathbf{Top}\) is evaluation at \(G/e\):

\[\Phi(\mathcal{F}) = \mathcal{F}(G/e).\]

The space \(\mathcal{F}(G/e)\) acquires a \(G\)-action via the \(G\)-action on \(G/e \cong G\) — morphisms in \(\mathcal{O}_G\) from \(G/e\) to itself correspond to elements of \(G\) (since \(\mathrm{Map}_G(G/e, G/e) = (G/e)^e = G\)), and this gives \(\mathcal{F}(G/e)\) a \(G\)-action.

Proof sketch via the bar construction.

Define a cofibrant replacement functor \(\Phi: \mathrm{Fun}(\mathcal{O}_G^{\mathrm{op}}, \mathbf{Top}) \to G\mathbf{Top}\) by the two-sided bar construction:

\[\Phi(\mathcal{F}) = \bigl| B_\bullet(\mathcal{F}, \mathcal{O}_G, M) \bigr|,\]

where \(M: \mathcal{O}_G \to G\mathbf{Top}\) is the functor \(M(G/H) = G/H\) (realizing the orbits in \(G\mathbf{Top}\)), and the bar construction has:

• \(B_n(\mathcal{F}, \mathcal{O}_G, M) = \displaystyle\coprod_{G/H_0 \to \cdots \to G/H_n} \mathcal{F}(G/H_0) \times G/H_n\),

with simplicial face maps given by composition in \(\mathcal{O}_G\) and by the \(\mathcal{F}\) and \(M\) actions.

The key computation: \(\Phi(\mathcal{F})^H = |\mathcal{F}(G/H \times \Delta^\bullet)| \simeq \mathcal{F}(G/H)\) via the extra degeneracy argument: there is a contraction of the simplicial set of \(H\)-fixed-point simplices onto the \(0\)-simplices, given by the identity coset \(eH \in (G/H)^H\).

This shows \(\psi \circ \Phi \simeq \mathrm{id}\) (up to weak equivalence), establishing the Quillen equivalence. \(\square\)

[!TIP]- Solution to Exercise 16 A presheaf on \(\mathcal{O}_{C_2}^{\mathrm{op}}\) is determined by: two spaces \(A = \mathcal{F}(C_2/e)\) and \(B = \mathcal{F}(C_2/C_2)\), a map \(\rho: B \to A\) (the restriction along \(C_2/e \to C_2/C_2\)), and an involution \(\sigma: A \to A\) (coming from the nontrivial endomorphism of \(C_2/e\), i.e., the generator \(\tau \in C_2\) acting on \(C_2/e \cong C_2\)).

The constraint is that \(\rho\) maps into the \(\sigma\)-fixed points: \(\sigma \circ \rho = \rho\) (since \(B\) maps to the fixed part of \(A\)).

The corresponding \(C_2\)-space under \(\Phi\) is the space \(A\) with \(C_2\)-action given by \(\sigma\), and \(B \cong A^{C_2}\) (the fixed-point subspace). So a genuine \(C_2\)-space is precisely the data of a space \(A\) with a \(C_2\)-involution \(\sigma\) — but the presheaf perspective makes the fixed-point data \(B\) explicit.

5.4 The Infinity-Categorical Statement

The Quillen equivalence of Elmendorf upgrades to an \((\infty,1)\)-categorical equivalence.

Theorem 1.3.8 (Elmendorf, \(\infty\)-categorical form). There is an equivalence of \((\infty,1)\)-categories:

\[G\mathbf{Top} \simeq \mathrm{Fun}(\mathcal{O}_G^{\mathrm{op}}, \mathbf{Spaces}),\]

where \(\mathbf{Spaces}\) denotes the \((\infty,1)\)-category of spaces (Kan complexes / \(\infty\)-groupoids).

The \((\infty,1)\)-categorical statement is strictly stronger: it says not only that the homotopy categories are equivalent, but that all higher homotopical data (mapping spaces, homotopy coherent diagrams, etc.) are also equivalent.

5.5 Applications

Elmendorf’s theorem has several important structural consequences.

Families of Subgroups. A family \(\mathcal{F}\) of subgroups of \(G\) is a collection closed under conjugation and taking subgroups. Examples: - \(\mathcal{F} = \{\{e\}\}\): the trivial family (only the trivial subgroup). - \(\mathcal{F} = \mathbf{All}\): all closed subgroups. - \(\mathcal{F} = \mathbf{Fin}\): all finite subgroups (relevant for \(G = S^1\)).

Definition (Classifying G-Space for a Family). For a family \(\mathcal{F}\), the classifying G-space \(E\mathcal{F}\) is the unique G-space (up to genuine weak equivalence) with:

\[E\mathcal{F}^H \simeq \begin{cases} * & H \in \mathcal{F} \\ \emptyset & H \notin \mathcal{F}. \end{cases}\]

Via Elmendorf’s theorem, this is the presheaf \(\mathcal{O}_G^{\mathrm{op}} \to \mathbf{Spaces}\) sending \(G/H \mapsto *\) for \(H \in \mathcal{F}\) and \(G/H \mapsto \emptyset\) for \(H \notin \mathcal{F}\).

[!EXAMPLE]- The Universal Space \(EG\) Taking \(\mathcal{F} = \{\{e\}\}\): \(E\mathcal{F} = EG\) (the universal free \(G\)-space), which has \((EG)^H = \emptyset\) for \(H \neq \{e\}\) and \((EG)^{\{e\}} \simeq *\). This is the contractible total space of the universal \(G\)-bundle \(EG \to BG\).

[!TIP]- Solution to Exercise 17 (a) \((S^\infty)^{C_2}\): fixed points of \(x \mapsto -x\) on \(S^\infty\) would require \(x = -x\), i.e., \(x = 0\), but \(0 \notin S^\infty\). So \((S^\infty)^{C_2} = \emptyset\). The underlying space \(S^\infty\) is contractible (it is the colimit of \(S^n\) along the standard inclusions, and each map \(S^n \hookrightarrow S^{n+1}\) is nullhomotopic in \(S^{n+1}\)). So \((S^\infty)^e = S^\infty \simeq *\).

2. The \(C_2\)-CW structure on \(S^n\) with antipodal action (from Exercise 11) gives \(S^\infty\) a \(C_2\)-CW structure with all cells of type \(C_2/e\) (free). The orbit space is \(S^\infty/C_2 = \mathbb{RP}^\infty\), and since \(S^\infty\) is contractible with free \(C_2\)-action, this is indeed a model for \(BC_2 = K(C_2, 1)\).

G-Connected Components. Via Elmendorf’s theorem, the equivariant \(\pi_0\) of a G-space \(X\) is the presheaf \(G/H \mapsto \pi_0(X^H)\). This records how the connected components of the fixed-point sets vary with the subgroup.

Bredon Cohomology. The Eilenberg-Mac Lane G-spaces are constructed via Elmendorf’s theorem from presheaves of abelian groups on \(\mathcal{O}_G\).

Definition (Coefficient System). A Bredon coefficient system is a functor \(M: \mathcal{O}_G^{\mathrm{op}} \to \mathbf{Ab}\) (a presheaf of abelian groups on \(\mathcal{O}_G\)).

Via Elmendorf’s theorem, one constructs an Eilenberg-Mac Lane G-space \(K(M, n)\) with \(\pi_n^H(K(M,n)) = M(G/H)\) and all other equivariant homotopy groups trivial. Bredon cohomology is then defined by

\[H^n_G(X; M) = [X, K(M,n)]_G,\]

the group of equivariant homotopy classes of maps, and this recovers Bredon’s original (1967) cohomology theory of \(G\)-spaces.

References