Genuine G-Spectra

Table of Contents

• 1. Stabilization and the Need for Genuine Spectra
  • 1.1 Naive Stabilization and Its Failures
  • 1.2 Representation Spheres
  • 1.3 What Genuine Stabilization Fixes
• 2. G-Universes and Indexing
  • 2.1 The Indexing Universe
  • 2.2 Complete vs. Incomplete Universes
  • 2.3 Spaces of Isometric Embeddings
• 3. Lewis–May–Steinberger G-Spectra
  • 3.1 G-Prespectra
  • 3.2 The Omega-Spectrum Condition
  • 3.3 Spectrification
• 4. Orthogonal G-Spectra
  • 4.1 The Indexing Category
  • 4.2 Orthogonal G-Spectra as Functors
  • 4.3 The Smash Product via Day Convolution
  • 4.4 Comparison with Lewis–May–Steinberger
• 5. Naive vs. Genuine: Two Model Structures
  • 5.1 The Naive Model Structure
  • 5.2 The Genuine Model Structure
  • 5.3 Lewis’s Theorem: Inequivalence
  • 5.4 What the Genuine Theory Gains
  • 5.5 G-CW-Spectra
• 6. Homotopy Groups as Mackey Functors
  • 6.1 Definition of the Homotopy Mackey Functor
  • 6.2 Restriction and Transfer Maps
  • 6.3 The Mackey Axiom from the Double Coset Formula
  • 6.4 The C2 Example in Detail
• 7. Suspension Spectra and Basic Examples
  • 7.1 The Equivariant Suspension Spectrum
  • 7.2 The Sphere Spectrum and the Burnside Ring
  • 7.3 Equivariant Eilenberg–Mac Lane Spectra
  • 7.4 The tom Dieck Splitting
• 8. The Smash Product and Closed Structure
  • 8.1 Symmetric Monoidal Structure
  • 8.2 The Internal Hom and Spanier–Whitehead Duality
  • 8.3 The Suspension Isomorphism
  • 8.4 Cofiber Sequences
• 9. Change of Universe and Change of Group
  • 9.1 Change of Universe
  • 9.2 Change of Group: Restriction
  • 9.3 Induction and Coinduction
  • 9.4 The Multiplicative Norm
  • 9.5 Preview: Fixed-Point Functors
• References

1. Stabilization and the Need for Genuine Spectra 📐

1.1 Naive Stabilization and Its Failures

The passage from unstable to stable homotopy theory classically proceeds by formally inverting \(S^1\): the stable homotopy category is obtained from \(G\mathbf{Top}_*\) by inverting \(\Sigma = S^1 \wedge -\). This yields a well-defined triangulated category. In the equivariant setting, one can perform the same construction to get naive G-spectra — orthogonal spectra equipped with a \(G\)-action, where weak equivalences are detected only on the underlying non-equivariant homotopy type.

The problem is that this is too coarse. There are several structural theorems in equivariant homotopy theory that require inverting suspension by all representation spheres \(S^V\), not just \(S^1\):

1. RO(G)-graded homotopy groups. For a genuine G-spectrum \(E\), one wants to define \(\pi_V^H(E) = [S^V, E]^H\) for any real representation \(V\) of \(G\) and subgroup \(H \leq G\). This requires that \(S^V \wedge -\) is invertible in the homotopy category.

2. The Wirthmuller isomorphism. For a subgroup inclusion \(i: H \hookrightarrow G\), the functors \(i_!(E) = G_+ \wedge_H E\) (left adjoint to restriction) and \(i_* = F_H(G_+, -)\) (right adjoint) are not equivalent in the naive setting. Here \(G_+\) carries a \((G,H)\)-bispace structure: left \(G\)-action by left multiplication and right \(H\)-action by right multiplication; the balanced smash product \(G_+ \wedge_H E\) quotients by \((gh, e) \sim (g, he)\) for \(h \in H\). In the genuine setting, the Wirthmuller isomorphism gives \(i_! E \simeq i_* E \wedge S^L\) where \(L = L_{G/H}\) is the tangent representation of \(G/H\). This fails naively.

3. The Adams isomorphism. For a free \(H\)-spectrum \(X\), the Adams isomorphism \((G_+ \wedge_H X)^G \simeq X^H\) is a theorem about genuine spectra; it has no naive analogue.

4. Equivariant duality. Atiyah duality \(D(\Sigma^\infty_+ M) \simeq \Sigma^\infty_+ M^{-TM}\) for a compact \(G\)-manifold \(M\) requires the tangent bundle to contribute a genuine representation sphere twist.

1.2 Representation Spheres

Definition (Representation Sphere). Let \(V\) be a finite-dimensional real representation of \(G\) — a finite-dimensional real inner product space with a continuous \(G\)-action by isometries. The representation sphere \(S^V\) is the one-point compactification of \(V\):

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

topologized so that the neighborhoods of \(\infty\) are complements of compact sets. As a \(G\)-space, \(G\) acts on \(V^+\) by extending the action on \(V\) with \(g \cdot \infty = \infty\).

The formula \(S^{V \oplus W} \cong S^V \wedge S^W\) holds as \(G\)-spaces, making representation spheres multiplicative.

1.3 What Genuine Stabilization Fixes

A genuine G-spectrum is obtained by formally inverting \(S^V \wedge -\) for all finite-dimensional real representations \(V\) of \(G\) simultaneously. This means:

• For every \(V\), there is an adjoint pair \((\Sigma^V, \Omega^V) = (S^V \wedge -, F(S^V, -))\) on the homotopy category.
• The RO(G)-graded homotopy groups \(\underline{\pi}_V(E)\) are well-defined for all \(V \in \mathrm{RO}(G)\).
• Transfers exist in stable homotopy, making homotopy groups into Mackey Functors rather than just abelian groups.

2. G-Universes and Indexing 🔑

2.1 The Indexing Universe

The classical Lewis–May–Steinberger approach indexes a \(G\)-spectrum on the finite-dimensional sub-inner-product-spaces of a single large “universe.”

Definition (G-Universe). A \(G\)-universe is a countably-infinite-dimensional real inner product space \(\mathcal{U}\) equipped with a continuous isometric \(G\)-action, together with a countably infinite family of pairwise orthogonal copies of each finite-dimensional representation of \(G\) in \(\mathcal{U}\) (so \(\mathcal{U}\) is not itself finite-dimensional). We require:

1. (Completeness under stabilization) \(\mathcal{U}\) contains a copy of the trivial representation \(\mathbb{R}\) (equivalently, \(\mathbb{R}^\infty \subset \mathcal{U}\) as a trivial subrepresentation).
2. (Stability under direct sum) For any \(V, W \subset \mathcal{U}\) finite-dimensional, \(V \oplus W \subset \mathcal{U}\).
3. (Topological conditions) \(\mathcal{U} \cong \bigoplus_{V \in \mathrm{Irr}_\mathbb{R}(G)} V^{\oplus \infty}\) as a real \(G\)-inner product space, where the direct sum is over the countably many (for finite \(G\)) irreducible real representations of \(G\).

The trivial universe \(\mathcal{U}_{\mathrm{triv}} = \mathbb{R}^\infty\) carries the trivial \(G\)-action; it sees only the single representation \(\mathbb{R}\).

2.2 Complete vs. Incomplete Universes

Definition (Complete Universe). A \(G\)-universe \(\mathcal{U}\) is complete if for every irreducible real representation \(V\) of \(G\), there are infinitely many orthogonal copies of \(V\) in \(\mathcal{U}\). Equivalently, \(\mathcal{U}\) is complete if and only if every finite-dimensional real \(G\)-representation embeds isometrically into \(\mathcal{U}\).

For a finite group \(G\), the complete \(G\)-universe is unique up to \(G\)-equivariant isometry:

\[\mathcal{U} \cong \bigoplus_{[V] \in \mathrm{Irr}_\mathbb{R}(G)} V^{\oplus \infty}.\]

For \(G = C_2\) there are two real irreducibles (\(\mathbf{1}\) and \(\sigma\)), so \(\mathcal{U} = \mathbb{R}^\infty \oplus \mathbb{R}^\infty_\sigma\) where the second factor is the sign representation repeated.

2.3 Spaces of Isometric Embeddings

For finite-dimensional sub-inner-product-spaces \(V, W \subset \mathcal{U}\), let \(\mathcal{L}(V, W)\) denote the space of linear isometric embeddings \(V \hookrightarrow W\) (i.e., injective linear maps preserving the inner product). This is a topological space with the subspace topology from \(\mathrm{Hom}(V, W)\).

The group \(G\) acts on \(\mathcal{L}(V, W)\) by \(g \cdot \phi = g \circ \phi \circ g^{-1}\) (conjugation via the \(G\)-actions on \(V\) and \(W\)). When \(V\) and \(W\) are \(G\)-representations, the \(G\)-fixed points \(\mathcal{L}(V, W)^G\) are the \(G\)-equivariant isometric embeddings.

For \(V \subset W\), the orthogonal complement \(W \ominus V\) is the representation \(W \ominus V = \{w \in W : \langle w, v \rangle = 0 \text{ for all } v \in V\}\). The structure maps of an LMS spectrum will involve suspension by \(W \ominus V\).

3. Lewis–May–Steinberger G-Spectra 📐

3.1 G-Prespectra

Fix a complete \(G\)-universe \(\mathcal{U}\) and write \(\mathcal{V}(\mathcal{U})\) for the poset of finite-dimensional sub-inner-product-spaces of \(\mathcal{U}\), ordered by inclusion.

Definition (G-Prespectrum). A \(G\)-prespectrum \(D\) indexed on \(\mathcal{U}\) consists of: 1. For each \(V \in \mathcal{V}(\mathcal{U})\), a based \(G\)-space \(D_V\). 2. For each pair \(V \subset W\) in \(\mathcal{V}(\mathcal{U})\), a based \(G\)-equivariant structure map

\[\sigma_{V,W}: \Sigma^{W \ominus V} D_V = S^{W \ominus V} \wedge D_V \longrightarrow D_W,\]

satisfying the transitivity condition: for \(V \subset W \subset X\), the diagram

commutes. Here we use \(S^{X \ominus W} \wedge S^{W \ominus V} \cong S^{X \ominus V}\) from the direct sum decomposition \(X = W \oplus (X \ominus W) = V \oplus (W \ominus V) \oplus (X \ominus W)\).

Passing to adjoints, each structure map \(\sigma_{V,W}\) has an adjoint

\[\tilde{\sigma}_{V,W}: D_V \longrightarrow \Omega^{W \ominus V} D_W = \mathrm{Map}_*(S^{W \ominus V}, D_W).\]

\(G\)-maps of \(G\)-prespectra are natural transformations respecting the structure maps. The category of \(G\)-prespectra is denoted \(G\mathcal{P}(\mathcal{U})\).

3.2 The Omega-Spectrum Condition

Definition (G-Spectrum). A \(G\)-prespectrum \(E\) is a \(G\)-spectrum (or \(\Omega\)-\(G\)-spectrum) if for every \(V \subset W\) in \(\mathcal{V}(\mathcal{U})\), the adjoint structure map

\[\tilde{\sigma}_{V,W}: E_V \xrightarrow{\;\sim\;} \Omega^{W \ominus V} E_W\]

is a homeomorphism of based \(G\)-spaces.

The category \(G\mathcal{S}(\mathcal{U})\) of \(G\)-spectra is the full subcategory of \(G\mathcal{P}(\mathcal{U})\) on the \(\Omega\)-\(G\)-spectra.

3.3 Spectrification

The inclusion \(G\mathcal{S}(\mathcal{U}) \hookrightarrow G\mathcal{P}(\mathcal{U})\) has a left adjoint \(L\), the spectrification functor.

Construction (Spectrification). For a \(G\)-prespectrum \(D\), define \(LD\) by

\[(LD)_V = \mathrm{colim}_{V \subset W} \Omega^{W \ominus V} D_W,\]

where the colimit is taken over the directed system of inclusions \(V \subset W\) in \(\mathcal{V}(\mathcal{U})\). The adjoint structure maps of \(LD\) are induced by passing the colimit.

Proposition. \(L: G\mathcal{P}(\mathcal{U}) \to G\mathcal{S}(\mathcal{U})\) is left adjoint to the inclusion, and the unit \(\eta: D \to L(D)\) is an isomorphism whenever \(D\) is already a \(G\)-spectrum.

The suspension spectrum is then defined properly as \(\Sigma^\infty_+ X = L(X \mapsto S^V \wedge X_+)\) — applying spectrification to the suspension prespectrum.

4. Orthogonal G-Spectra 🔑

4.1 The Indexing Category

The modern approach, due to Mandell–May and developed in Schwede’s work, replaces the universe-indexed approach with a categorical definition that is more amenable to homotopy-theoretic analysis.

Definition (The Indexing Category \(\mathbf{I}_G\)). Let \(\mathbf{I}_G\) be the symmetric monoidal topological category defined as follows: - Objects: finite-dimensional real inner product spaces \(V\) (not necessarily \(G\)-representations — the \(G\)-action enters through the morphism spaces). - Morphisms: \(\mathbf{I}_G(V, W)\) is the Thom space of the tautological vector bundle \(\gamma_{V,W}\) over the space of isometric embeddings \(\mathrm{Inj}(V, W)\):

\[\mathbf{I}_G(V, W) = \mathrm{Th}(\gamma_{V,W}) = \mathrm{Inj}(V, W)^+ \wedge_{O(\dim V)} S^{\dim W - \dim V}\]

when \(\dim V \leq \dim W\), and \(\mathbf{I}_G(V, W) = *\) otherwise. Here \(\gamma_{V,W}\) is the bundle over \(\mathrm{Inj}(V, W)\) whose fiber at \(\phi: V \hookrightarrow W\) is the orthogonal complement \(\phi(V)^\perp \subset W\).

• G-action on morphisms: \(G\) acts on \(\mathbf{I}_G(V, W)\) through a representation structure on \(V\) and \(W\) (when we evaluate an orthogonal \(G\)-spectrum on a \(G\)-representation \(V\), \(G\) acts on \(V\) by the representation and thus on morphism spaces by conjugation).

• Monoidal structure: \(\oplus\) on objects, and the monoidal structure on morphisms defined by the smash product of Thom spaces.

4.2 Orthogonal G-Spectra as Functors

Definition (Orthogonal G-Spectrum). An orthogonal \(G\)-spectrum is a based continuous functor

\[X: \mathbf{I}_G \longrightarrow G\mathbf{Top}_*\]

from the indexing category to based \(G\)-spaces. Explicitly: - For each finite-dimensional real inner product space \(V\), a based \(G\)-space \(X(V)\). - For each pair \(V, W\), a based continuous structure map

\[\sigma_{V,W}: \mathbf{I}_G(V, W) \wedge X(V) \longrightarrow X(W)\]

natural in \(V\) and \(W\), and satisfying the evident associativity and unitality conditions.

The \(G\)-action on \(X(V)\) for a \(G\)-representation \(V\): when \(V\) carries a \(G\)-representation structure, \(G\) acts on \(X(V)\) via the induced action. An element \(g \in G\) gives an isometry \(g: V \to V\) (by the representation), which is a morphism in \(\mathbf{I}_G(V, V)\); the structure map then gives a self-map \(X(g): X(V) \to X(V)\).

The category of orthogonal \(G\)-spectra is denoted \(\mathrm{Sp}^G_O\) or \(G\mathcal{S}\).

4.3 The Smash Product via Day Convolution

The smash product of orthogonal \(G\)-spectra is defined via Day convolution with respect to the monoidal structure \(\oplus\) on \(\mathbf{I}_G\).

Definition (Smash Product). For orthogonal \(G\)-spectra \(X\) and \(Y\), define their smash product \(X \wedge Y\) by the coend formula:

\[(X \wedge Y)(V) = \int^{(A, B) \in \mathbf{I}_G \times \mathbf{I}_G} \mathbf{I}_G(A \oplus B, V) \wedge X(A) \wedge Y(B).\]

Equivalently, \(X \wedge Y\) is the left Kan extension of the external smash product \((V, W) \mapsto X(V) \wedge Y(W)\) along the direct sum functor \(\oplus: \mathbf{I}_G \times \mathbf{I}_G \to \mathbf{I}_G\):

\[X \wedge Y = \mathrm{Lan}_\oplus (X \boxtimes Y).\]

Theorem (Mandell–May). The smash product makes \(\mathrm{Sp}^G_O\) into a closed symmetric monoidal category:

\[(\mathrm{Sp}^G_O, \wedge, \mathbb{S})\]

where \(\mathbb{S} = \Sigma^\infty_+ \{*\}\) is the sphere spectrum. The unit and associativity isomorphisms are strictly coherent (not just up to homotopy), making this a genuine symmetric monoidal structure on the point-set level.

4.4 Comparison with Lewis–May–Steinberger

Theorem (Mandell–May). There is a Quillen equivalence of model categories

\[\mathrm{Sp}^G_O \simeq_Q G\mathcal{S}(\mathcal{U})\]

between orthogonal \(G\)-spectra with the genuine model structure and Lewis–May–Steinberger \(G\)-spectra indexed on a complete \(G\)-universe \(\mathcal{U}\). The equivalence is given by evaluation: send \(X \in \mathrm{Sp}^G_O\) to the LMS spectrum \(V \mapsto X(V)\) for \(V \subset \mathcal{U}\).

This means that the two models present the same homotopy theory. The orthogonal model is preferred for: - Having a strict smash product. - Being amenable to global equivariant homotopy theory (varying \(G\)). - Having explicit cofibrant/fibrant replacement functors.

5. Naive vs. Genuine: Two Model Structures ⚠️

5.1 The Naive Model Structure

On the category \(\mathrm{Sp}^G_O\) of orthogonal \(G\)-spectra, there is a Quillen model structure with:

• Naive weak equivalences: \(f: X \to Y\) is a naive weak equivalence if the underlying map of non-equivariant orthogonal spectra \(f^e: X^e \to Y^e\) (forgetting the \(G\)-action, i.e., evaluating at the trivial subgroup \(e \leq G\)) is a \(\pi_*\)-isomorphism.
• Naive fibrations: the fibrations and cofibrations are defined by the projective model structure — they are obtained by forgetting the \(G\)-action.

The homotopy category of the naive model structure is denoted \(\mathrm{Ho}^G_{\mathrm{naive}}(\mathrm{Sp}^G_O)\).

5.2 The Genuine Model Structure

Definition (Genuine Weak Equivalence). A map \(f: X \to Y\) of orthogonal \(G\)-spectra is a genuine \(G\)-weak equivalence if for every closed subgroup \(H \leq G\) and every integer \(n \in \mathbb{Z}\), the induced map

\[\pi_n^H(f): \pi_n^H(X) \longrightarrow \pi_n^H(Y)\]

is an isomorphism, where \(\pi_n^H(X) = [S^n, X(V)^H]\) for large enough representation \(V\) containing \(\mathbb{R}^n\) (stabilized via the structure maps).

More precisely, for a representation sphere \(S^V\) with \(V\) a \(G\)-representation: \[\pi_n^H(X) = \mathrm{colim}_{V \subset \mathcal{U}} [S^{n + V}, X(V)]^H\] where \([-, -]^H\) denotes homotopy classes of \(H\)-equivariant based maps and the colimit is over \(V\) ranging over finite-dimensional sub-representations of the complete \(G\)-universe.

The genuine model structure has: - Weak equivalences: genuine \(G\)-weak equivalences (as defined above). - Fibrations: maps \(f: X \to Y\) such that \(f^H: X^H \to Y^H\) is an \(\Omega\)-spectrum level fibration for all \(H \leq G\). - Cofibrations: determined by the left-lifting property.

Theorem (Mandell–May). The genuine model structure on \(\mathrm{Sp}^G_O\) is a proper, cellular, stable model category. The smash product \(\wedge\) is a Quillen bifunctor with respect to this model structure, making \((\mathrm{Sp}^G_O, \wedge, \mathbb{S})\) into a symmetric monoidal model category.

5.3 Lewis’s Theorem: Inequivalence

The key structural theorem distinguishing naive from genuine is:

Theorem (Lewis). If \(G \neq \{e\}\), the naive and genuine equivariant stable homotopy categories are not equivalent. More precisely, the forgetful functor

\[\iota^*: \mathrm{Ho}^G_{\mathrm{genuine}}(\mathrm{Sp}^G_O) \longrightarrow \mathrm{Ho}^G_{\mathrm{naive}}(\mathrm{Sp}^G_O)\]

does not induce an equivalence of triangulated categories.

Proof sketch. The sphere spectrum \(\mathbb{S}\) has

\[\underline{\pi}_0^{\mathrm{genuine}}(\mathbb{S}) = \underline{A}(G), \quad \underline{\pi}_0^{\mathrm{naive}}(\mathbb{S}) = \underline{\mathbb{Z}},\]

where \(\underline{A}(G)\) is the Burnside ring Mackey functor (with \(\underline{A}(G)(G/H) = A(H)\), the Burnside ring of \(H\), including transfer maps) and \(\underline{\mathbb{Z}}\) is the constant Mackey functor. Since \(A(H) \neq \mathbb{Z}\) for \(H \neq \{e\}\) (the Burnside ring has additional generators from non-trivial \(H\)-sets), these two objects are not isomorphic as Mackey functors. Therefore \(\mathbb{S}^{\mathrm{genuine}} \not\simeq \mathbb{S}^{\mathrm{naive}}\) in any putative equivalence. \(\square\)

5.4 What the Genuine Theory Gains

flowchart LR
    subgraph Unstable
        GTop["Genuine G-spaces
weq: all X^H equiv
Elmendorf: PSh(O_G)"]
        NTop["Naive/Borel G-spaces
weq: underlying X equiv
Fun(BG, Top)"]
    end
    subgraph Stable
        GSp["Genuine G-spectra
spectral Mackey functors
pi_* = Mackey functors"]
        NSp["Naive = Borel G-spectra
Fun(BG, Sp)
pi_* = plain groups with G-action"]
    end
    GTop -->|"forget fixed-pt weq"| NTop
    GTop -->|"Sigma^inf_+"| GSp
    NTop -->|"Sigma^inf_+"| NSp
    GSp -->|"iota* (trivial universe)"| NSp

5.5 G-CW-Spectra 📐

The model structure language above has a classical CW-theoretic counterpart: just as G-spaces are built from orbit cells \(G/H \times D^n\), genuine G-spectra are built from representation-disk cells.

Definition (Naive G-CW-Spectrum). A naive G-CW-spectrum is a sequential colimit

\[* = X_{-1} \hookrightarrow X_0 \hookrightarrow X_1 \hookrightarrow \cdots\]

where each \(X_n\) is obtained from \(X_{n-1}\) by attaching cells of the form \(G/H_+ \wedge D^n_+\) along attaching maps \(G/H_+ \wedge S^{n-1} \to X_{n-1}\). Here \(H \leq G\) is any subgroup and \(D^n, S^{n-1}\) carry trivial \(G\)-action. These are exactly the generating cofibrations of the naive model structure:

\[I_{\mathrm{naive}} = \{ G/H_+ \wedge (S^{n-1} \hookrightarrow D^n) : H \leq G,\, n \geq 0 \}.\]

Definition (Genuine G-CW-Spectrum). A genuine G-CW-spectrum is built by attaching representation-disk cells. For a finite-dimensional \(G\)-representation \(V\), define the representation disk \(D^V = \{v \in V : |v| \leq 1\}\) and representation sphere \(S^{V-1} = \partial D^V\). The generating cofibrations of the genuine model structure are:

\[I_{\mathrm{gen}} = \{ G/H_+ \wedge (S^{V-1} \hookrightarrow D^V) : H \leq G,\, V \in \mathcal{U}^H \},\]

where \(\mathcal{U}^H\) denotes the finite-dimensional \(H\)-fixed sub-representations of the complete universe. A genuine G-CW-spectrum is a sequential colimit obtained by attaching cells of this type.

The Equivariant Whitehead Theorem. The equivariant analogue of the classical Whitehead theorem holds for genuine G-CW-spectra:

Theorem (Whitehead). Let \(f: X \to Y\) be a map of genuine G-CW-spectra. If \(\pi_n^H(f)\) is an isomorphism for all \(H \leq G\) and all \(n \in \mathbb{Z}\), then \(f\) is a \(G\)-homotopy equivalence.

Proof sketch. By the model structure, \(f\) is a genuine weak equivalence between cofibrant objects, hence a \(G\)-homotopy equivalence by Ken Brown’s lemma. \(\square\)

Cells, slices, and the filtration. The generating cofibrations \(I_{\mathrm{gen}}\) refine to a slice-graded family. Define the slice dimension of a cell \(G/H_+ \wedge D^V\) to be \(\dim_\mathbb{R}(V)\). The slice cells

\[\widetilde{S}^{m|H|} = G_+ \wedge_H S^{m\rho_H}\]

of Equivariant Postnikov and Slice are specific genuine G-CW cells built from the regular representation \(\rho_H\). The slice filtration on a genuine G-CW-spectrum \(E\) is the sub-CW-filtration generated by attaching cells of slice dimension \(\leq n\):

\[\text{slice filtration} \;\subset\; \text{genuine G-CW filtration} \;\subset\; RO(G)\text{-graded cell filtration.}\]

The slice filtration is coarser: it groups all cells of the same real dimension together, regardless of which representation they come from.

Compatibility with change-of-group. Induction and restriction act on G-CW-spectra in the expected way: - Restriction \(i_H^*: \mathrm{Sp}^G \to \mathrm{Sp}^H\) sends a genuine G-CW-spectrum to a genuine H-CW-spectrum. A cell \(G/K_+ \wedge D^V\) restricts to \(\bigsqcup_{[g] \in H \backslash G / K} H/(H \cap {}^gK)_+ \wedge D^{V|_{H \cap {}^gK}}\) by the Mackey double coset formula — each double coset contributes one \(H\)-cell. - Induction \(G_+ \wedge_H -: \mathrm{Sp}^H \to \mathrm{Sp}^G\) sends an \(H\)-cell \(H/K_+ \wedge D^V\) to a \(G\)-cell \(G/K_+ \wedge D^V\).

6. Homotopy Groups as Mackey Functors 🔑

6.1 Definition of the Homotopy Mackey Functor

For a genuine \(G\)-spectrum \(E \in \mathrm{Sp}^G_O\), the equivariant homotopy groups assemble into a Mackey Functor \(\underline{\pi}_n(E)\).

Definition (Homotopy Mackey Functor). For \(n \in \mathbb{Z}\) and \(E \in \mathrm{Sp}^G_O\), define the \(n\)-th homotopy Mackey functor \(\underline{\pi}_n(E)\) by

\[\underline{\pi}_n(E)(G/H) = \pi_n(E^H) = \mathrm{colim}_{V \subset \mathcal{U}} [S^{n + V}, E(V)]^H,\]

where \(E^H\) denotes the \(H\)-fixed-point spectrum. The colimit is over finite-dimensional \(G\)-sub-representations \(V \subset \mathcal{U}\) (with \(\dim V \geq \max(0, -n)\)), and a map \(S^{n+V} \to E(V)\) is required to be \(H\)-equivariant.

The value \(\underline{\pi}_n(E)(G/H) = \pi_n^H(E)\) is simply the \(n\)-th homotopy group of the \(H\)-fixed-point space of \(E\) at a sufficiently large representation.

6.2 Restriction and Transfer Maps

To complete the Mackey functor structure, we need maps in both directions.

Restriction maps. For an inclusion of subgroups \(i: K \hookrightarrow H\), the map \(G/K \to G/H\) defined by \(gK \mapsto gH\) is a morphism in the orbit category. The restriction

\[\mathrm{res}^H_K: \pi_n^H(E) \longrightarrow \pi_n^K(E)\]

is simply the map induced by the inclusion of fixed-point spaces \(E^H \hookrightarrow E^K\) (a smaller group has more fixed points): if \(f: S^{n+V} \to E(V)\) is \(H\)-equivariant, it is also \(K\)-equivariant since \(K \leq H\).

Transfer maps. For \(K \leq H \leq G\), the transfer is the stable map

\[\mathrm{tr}^H_K: \pi_n^K(E) \longrightarrow \pi_n^H(E)\]

constructed via the Pontryagin–Thom collapse. Specifically, the coset space \(H/K\) is a finite \(H\)-set, and there is a stable transfer map \(\mathrm{tr}: \Sigma^\infty_+ (H/K) \to \mathbb{S}^H\) in the \(H\)-equivariant stable category. The transfer on homotopy groups is then the composition

\[\pi_n^K(E) \cong [S^{n}, E^K] \xrightarrow{\mathrm{tr}^H_K \wedge -} [\Sigma^\infty_+(H/K) \wedge S^n, \Sigma^\infty_+(H/K) \wedge E^K] \to [S^n, E^H].\]

Conjugation maps. For \(g \in G\) and \(H \leq G\), conjugation by \(g\) gives a group isomorphism \(c_g: H \to {}^gH = gHg^{-1}\), and this induces an isomorphism

\[c_g^*: \pi_n^{{}^gH}(E) \xrightarrow{\;\sim\;} \pi_n^H(E).\]

6.3 The Mackey Axiom from the Double Coset Formula

We must verify that the Mackey double coset formula holds: for \(K, L \leq H\),

\[\mathrm{res}^H_K \circ \mathrm{tr}^H_L = \sum_{[h] \in K \backslash H / L} \mathrm{tr}^K_{K \cap {}^hL} \circ c_{h}^* \circ \mathrm{res}^L_{K^h \cap L}.\]

In the stable equivariant category, this follows from the geometric decomposition of the span \(H/K \xleftarrow{} H \xrightarrow{} H/L\) as a disjoint union of double coset spans. Specifically, the fiber product

\[(H/K) \times_{H/e} (H/L) \cong \bigsqcup_{[h] \in K \backslash H / L} K/(K \cap {}^hL)\]

as \(K\)-sets, and the Mackey formula is the image of this identity under the stable transfer. This is a direct consequence of the Mackey double coset formula for finite \(G\)-sets (see Mackey Functors §4 for the algebraic statement).

Corollary. For any genuine \(G\)-spectrum \(E\) and any \(n \in \mathbb{Z}\), the data \((\underline{\pi}_n(E), \mathrm{res}, \mathrm{tr}, c_g)\) form a Mackey functor. The assignment \(E \mapsto \underline{\pi}_n(E)\) is a functor \(\mathrm{Ho}(\mathrm{Sp}^G_O) \to \mathbf{Mack}(G)\).

6.4 The C2 Example in Detail

Let \(G = C_2 = \{1, \sigma\}\). A \(C_2\)-Mackey functor is a diagram

\[M(C_2/C_2) = M^{C_2} \underset{\mathrm{tr}}{\overset{\mathrm{res}}{\rightleftharpoons}} M(C_2/e) = M^e\]

of abelian groups with \(\mathrm{res}: M^{C_2} \to M^e\) and \(\mathrm{tr}: M^e \to M^{C_2}\), satisfying the Mackey axiom \(\mathrm{tr} \circ \mathrm{res} = 1 + \sigma_*\) (where \(\sigma_*\) is conjugation by \(\sigma\), which acts as the identity on \(M^{C_2}\) and by the \(C_2\)-action on \(M^e\)).

For a genuine \(C_2\)-spectrum \(E\):

\[\underline{\pi}_n(E): \quad \pi_n(E^{C_2}) \underset{\mathrm{tr}}{\overset{\mathrm{res}}{\rightleftharpoons}} \pi_n(E^e)\]

• \(\mathrm{res} = i^*\): the map induced by the inclusion of fixed points \(E^{C_2} \hookrightarrow E^e\) (which simply forgets the \(C_2\)-equivariance condition).
• \(\mathrm{tr}\): the stable transfer, which for \(C_2/e\) can be described as follows — embed \(S^n \hookrightarrow S^n \vee S^n \cong C_2 \cdot S^n\) and apply the fold map; the stable transfer is the composition of this with the \(C_2\)-norm.
• Weyl conjugation: \(W_{C_2}(e) = C_2\) acts on \(\pi_n(E^e)\) by the residual \(C_2\)-action on the underlying spectrum. The Mackey axiom gives \(\mathrm{tr} \circ \mathrm{res} = 1 + \sigma_*\).

7. Suspension Spectra and Basic Examples 💡

7.1 The Equivariant Suspension Spectrum

Definition (Equivariant Suspension Spectrum). For a based \(G\)-space \(X\), the equivariant suspension spectrum is the orthogonal \(G\)-spectrum

\[\Sigma^\infty X \in \mathrm{Sp}^G_O, \quad (\Sigma^\infty X)(V) = S^V \wedge X,\]

with structure maps \(\mathbf{I}_G(V, W) \wedge S^V \wedge X \to S^W \wedge X\) induced by the tautological map \(\mathbf{I}_G(V, W) \wedge S^V \to S^W\).

For an unbased \(G\)-space \(Y\), define \(\Sigma^\infty_+ Y = \Sigma^\infty(Y_+)\) where \(Y_+ = Y \sqcup \{*\}\) adjoins a disjoint basepoint.

The infinite loop space functor \(\Omega^\infty: \mathrm{Sp}^G_O \to G\mathbf{Top}_*\) is defined by \(\Omega^\infty E = \mathrm{colim}_{V \subset \mathcal{U}} \Omega^V E(V)\).

Proposition (Suspension-Loop Adjunction). There is a natural adjunction

\[\Sigma^\infty_+: G\mathbf{Top} \rightleftharpoons \mathrm{Sp}^G_O : \Omega^\infty\]

with \(\Sigma^\infty_+\) left adjoint to \(\Omega^\infty\). The unit \(\eta: Y_+ \to \Omega^\infty \Sigma^\infty_+ Y\) and counit \(\epsilon: \Sigma^\infty_+ \Omega^\infty E \to E\) are the standard suspension-loop unit and counit.

7.2 The Sphere Spectrum and the Burnside Ring

Definition (Sphere Spectrum). The equivariant sphere spectrum is

\[\mathbb{S} = \Sigma^\infty_+ \{*\} = \Sigma^\infty S^0.\]

As an orthogonal \(G\)-spectrum, \(\mathbb{S}(V) = S^V\) with the canonical structure maps. The \(H\)-fixed points of \(\mathbb{S}\) at each level are \((\mathbb{S}(V))^H = (S^V)^H = S^{V^H}\), where \(V^H\) is the fixed subrepresentation.

Definition (Representation Sphere Spectrum). For a \(G\)-representation \(V\), define the representation sphere spectrum

\[S^V = \Sigma^\infty S^V \in \mathrm{Sp}^G_O.\]

More precisely, \(S^V\) is the suspension spectrum of the representation sphere, and it represents the functor \(\pi_V^H(E) = [S^V, E]^H\) in the stable category.

The suspension spectrum of the orbit \(G/H\) plays a central role: \(\Sigma^\infty_+ G/H\) is free as a \(G\)-spectrum (in the sense that its \(H\)-equivariant homotopy groups are those of a free \(G\)-space).

Proposition. \(\underline{\pi}_0(\mathbb{S}) = \underline{A}(G)\), the Burnside ring Mackey functor with \(\underline{A}(G)(G/H) = A(H)\) (the Burnside ring of \(H\)).

Sketch. The group \(\pi_0^H(\mathbb{S}) = [S^0, S^0]^H_{\mathrm{stable}} \cong A(H)\) by the equivariant stable Freudenthal theorem: stable maps \(S^0 \to S^0\) in the \(H\)-equivariant stable category biject with virtual finite \(H\)-sets, giving the Burnside ring. See Mackey Functors §5.2 for the Burnside ring Mackey functor.

7.3 Equivariant Eilenberg–Mac Lane Spectra

For a Mackey Functor \(\underline{M}\), there is an equivariant Eilenberg–Mac Lane spectrum \(H\underline{M}\) characterized by

\[\underline{\pi}_n(H\underline{M}) = \begin{cases} \underline{M} & n = 0, \\ 0 & n \neq 0. \end{cases}\]

Construction. One constructs \(H\underline{M}\) via the equivariant bar construction: for each \(G/H\), the \(H\)-fixed-point space of \(H\underline{M}\) is the classical Eilenberg–Mac Lane space \(K(\underline{M}(G/H), 0)\), and the equivariant structure is assembled using the restriction and transfer maps of \(\underline{M}\).

The spectrum \(H\underline{M}\) represents Bredon cohomology: for a \(G\)-CW complex \(X\),

\[H^n_G(X; \underline{M}) = [X, \Sigma^n H\underline{M}]^G.\]

7.4 The tom Dieck Splitting

One of the most important structural results for suspension spectra is the tom Dieck splitting.

Theorem (tom Dieck Splitting). For a based \(G\)-space \(X\) and a finite group \(G\),

\[\pi_n^G(\Sigma^\infty_+ X) \cong \bigoplus_{[H] \leq G} \pi_n^{\mathrm{st}}(EW_GH_+ \wedge_{W_GH} X^H),\]

where the sum is over conjugacy classes of subgroups, \(W_GH = N_GH / H\) is the Weyl group, and \(\pi_n^{\mathrm{st}}\) denotes stable homotopy groups.

Proof sketch. Use the \(H\)-fixed-point decomposition of the Pontryagin–Thom construction for free orbits. Each summand corresponds to the contribution from \(H\)-fixed locus of \(X\), twisted by the Weyl group \(W_GH\) action on \(EW_GH \times X^H\). \(\square\)

8. The Smash Product and Closed Structure 📐

8.1 Symmetric Monoidal Structure

We have already constructed the smash product via Day convolution in §4.3. Here we record its properties.

Theorem. The homotopy category \(\mathrm{Ho}(\mathrm{Sp}^G_O)\) with the derived smash product \(\wedge^L\) is a closed symmetric monoidal category:

\[(\mathrm{Ho}(\mathrm{Sp}^G_O), \wedge^L, \mathbb{S}).\]

The monoidal structure satisfies: 1. (Unit) \(\mathbb{S} \wedge E \simeq E \simeq E \wedge \mathbb{S}\). 2. (Associativity) \((E \wedge F) \wedge K \simeq E \wedge (F \wedge K)\) naturally. 3. (Commutativity) \(E \wedge F \simeq F \wedge E\) via the twist. 4. (Closed) There exists an internal hom \(F(E, F)\) with natural adjunction isomorphism \([E \wedge F, K]^G \cong [E, F(F, K)]^G\).

8.2 The Internal Hom and Spanier–Whitehead Duality

Definition (Internal Hom). For orthogonal \(G\)-spectra \(E, F\), the function spectrum \(F(E, F)\) is defined by

\[F(E, F)(V) = \mathrm{Map}_*(E(V), F(V))^G\]

(maps of \(G\)-spaces, assembled using the structure of orthogonal spectra). More precisely, \(F(E, F)\) represents the functor \(X \mapsto [X \wedge E, F]^G\) on the homotopy category.

Definition (Spanier–Whitehead Dual). The Spanier–Whitehead dual of \(E \in \mathrm{Sp}^G_O\) is

\[D(E) = F(E, \mathbb{S}).\]

A \(G\)-spectrum \(E\) is dualizable if the natural map \(E \wedge D(E) \to F(D(E), E)\) is an equivalence, or equivalently, if the evaluation map \(D(E) \wedge E \to \mathbb{S}\) and coevaluation \(\mathbb{S} \to E \wedge D(E)\) exhibit the standard adjunction.

Theorem (Equivariant Atiyah Duality). For a compact smooth \(G\)-manifold \(M\) with \(G\)-equivariant embedding \(M \hookrightarrow V\) into a \(G\)-representation \(V\), the equivariant Spanier–Whitehead dual is

\[D(\Sigma^\infty_+ M) \simeq \Sigma^\infty_+(M^{-TM}) = \Sigma^{-V} \Sigma^\infty(M^{\nu})\]

where \(\nu\) is the normal bundle of the embedding and \(M^\nu\) is the Thom space.

8.3 The Suspension Isomorphism

Proposition (Suspension Isomorphism). For any \(G\)-representation \(V\) and any \(G\)-spectrum \(E\), there is a natural equivalence

\[\Sigma^V E := S^V \wedge E \simeq E[V]\]

in the homotopy category, where \(E[V]\) is the \(V\)-th shift of \(E\). In particular: - \(\pi_n^H(\Sigma^V E) \cong \pi_{n + \dim V^H}^H(E)\) for the real dimension of the \(H\)-fixed subrepresentation. - More precisely (for the genuine RO(G)-graded story), \(\underline{\pi}_{V+n}(E)(G/H) \cong \underline{\pi}_n(E)(G/H)\) for a \(G\)-representation \(V\) thought of as an element of \(\mathrm{RO}(G)\).

The suspension isomorphism demonstrates that genuine \(G\)-spectra are stable with respect to all representation spheres: inverting \(S^V \wedge -\) for all \(V \in \mathrm{RO}(G)\) is the correct notion of equivariant stability.

8.4 Cofiber Sequences

A cofiber sequence in \(\mathrm{Sp}^G_O\) is a sequence \(A \xrightarrow{f} X \to C(f)\) where \(C(f) = X \cup_f \mathrm{cone}(A) = X \cup_f CA\) is the mapping cone. In the stable category:

• Every cofiber sequence \(A \to X \to X/A\) extends to a long exact sequence in equivariant homotopy groups:

\[\cdots \to \underline{\pi}_{n+1}(X/A) \to \underline{\pi}_n(A) \to \underline{\pi}_n(X) \to \underline{\pi}_n(X/A) \to \cdots\]

where all maps are morphisms of Mackey functors (not just abelian groups).

• The stable category is triangulated: the cofiber of the cofiber is the suspension: \(C(C(f)) \simeq \Sigma A\).

• Every cofiber sequence gives a fiber sequence via \(\Omega^\infty\): \(F(f) \simeq \Omega C(f)\) in the stable category.

9. Change of Universe and Change of Group 🔑

9.1 Change of Universe

Given an inclusion of \(G\)-universes \(\iota: \mathcal{U}' \hookrightarrow \mathcal{U}\), there are adjoint functors:

Definition (Change-of-Universe). The restriction functor \(\iota^*: \mathrm{Sp}^G(\mathcal{U}) \to \mathrm{Sp}^G(\mathcal{U}')\) is defined by \((\iota^* E)_V = E_V\) for \(V \subset \mathcal{U}'\) (restricting the indexing to sub-representations of \(\mathcal{U}'\)). Its left adjoint \(\iota_*\) (extension of universe) is defined by

\[(\iota_* D)_W = \mathrm{colim}_{V \subset \mathcal{U}', V \subset W} \Omega^{W \ominus V} D_V.\]

The adjunction \((\iota_*, \iota^*)\) is a Quillen adjunction but not a Quillen equivalence unless \(\mathcal{U} = \mathcal{U}'\).

9.2 Change of Group: Restriction

For a closed subgroup inclusion \(i: H \hookrightarrow G\), there is a restriction functor on spectra.

Definition (Restriction). The restriction functor \(i^* = \mathrm{res}^G_H: \mathrm{Sp}^G \to \mathrm{Sp}^H\) is defined by restricting the \(G\)-action to \(H\):

\[(i^* E)(V) = E(V)|_H,\]

where \(V\) is now regarded as an \(H\)-representation (by restriction). On homotopy groups, \(i^*\) sends \(\underline{\pi}_n(E)(G/K) = \pi_n^K(E)\) to \(\underline{\pi}_n(i^*E)(H/K \cap H) = \pi_n^{K \cap H}(i^*E)\).

9.3 Induction and Coinduction

The restriction functor has both a left and right adjoint.

Definition (Induction). The induction functor \(G_+ \wedge_H -: \mathrm{Sp}^H \to \mathrm{Sp}^G\) is

\[(G_+ \wedge_H E)(V) = G_+ \wedge_H E(V),\]

where \(G_+ \wedge_H X = (G \times X) / \{(gh, x) \sim (g, hx)\}\) is the balanced product. This is the left adjoint to restriction:

\[[G_+ \wedge_H E, F]^G \cong [E, i^* F]^H.\]

Definition (Coinduction). The coinduction functor \(F_H(G_+, -): \mathrm{Sp}^H \to \mathrm{Sp}^G\) is

\[F_H(G_+, E)(V) = F_H(G_+, E(V))= \mathrm{Map}_H(G, E(V)),\]

the \(H\)-equivariant maps from \(G_+\) to \(E(V)\). This is the right adjoint to restriction:

\[[i^* F, E]^H \cong [F, F_H(G_+, E)]^G.\]

Theorem (Wirthmuller Isomorphism). For a finite group \(G\) and \(H \leq G\), there is a natural equivalence of \(G\)-spectra

\[G_+ \wedge_H E \simeq F_H(G_+, E) \otimes S^{L_{G/H}},\]

where \(L_{G/H}\) is the tangent representation of \(G/H\) at the identity coset (the adjoint representation of \(H\) in \(G\)). In particular, induction \(\simeq\) coinduction up to a representation sphere twist.

flowchart LR
    A["Sp^H"] -->|"G_+ wedge_H -
(induction)"| B["Sp^G"]
    B -->|"i^* = res^G_H
(restriction)"| A
    A -->|"F_H(G_+, -)
(coinduction)"| B
    style A fill:#d4e6f1
    style B fill:#d5f5e3

9.4 The Multiplicative Norm

The induction functor \(G_+ \wedge_H -\) of §9.3 is additive: it sends wedges to wedges. The equivariant stable category supports a second, multiplicative induction \(N_H^G: \mathrm{Sp}^H \to \mathrm{Sp}^G\), called the norm, which sends smash products to smash products and is the left adjoint to restriction on commutative ring spectra. Its construction is one of the key innovations of Hill–Hopkins–Ravenel.

The categorical foundation: \(\mathrm{Sp}^G_O \cong S^{BG}\)

The norm is made possible by a fundamental equivalence of categories. Define the indexing category \(J\) (for orthogonal spectra) as the category of finite-dimensional real inner product spaces with linear isometric embeddings, and \(J^G\) as the full subcategory on finite-dimensional real \(G\)-representations. Let \(i: J \hookrightarrow J^G\) be the inclusion of trivially-indexed spaces.

Proposition A.12 (HHR, §A.2.3). The forgetful functor gives an equivalence of topological G-categories: \(J^G \simeq\) (objects of \(J\) with \(G\)-action).

The proof turns on Lemma A.20 (Mandell–May, Lemma V.1.5): for any two \(G\)-representations \(V, W\) of the same dimension,

\[O(V, U) \times_{O(V)} O(W, V) \cong O(W, U)\]

as a coequalizer in \(T^G\). Since any \(G\)-representation \(W\) of dimension \(n\) is non-equivariantly isomorphic to \(\mathbb{R}^n\), the coequalizer shows that the representation sphere \(S^{-W}\) is isomorphic to the left Kan extension \(i_!(S^{-\mathbb{R}^n})\). Everything in \(S^G\) is therefore determined by the trivially-indexed levels.

Proposition A.19 (HHR, §A.2.6). The restriction and left Kan extension functors

\[i^*: \mathrm{Cat}^G(J^G, T^G) \rightleftharpoons \mathrm{Cat}^G(J, T^G) :i_!\]

are inverse equivalences of symmetric monoidal categories. In other words, \(\mathrm{Sp}^G_O\) is equivalent to the category \(S^{BG}\) of orthogonal spectra equipped with a \(G\)-action.

HHR flag the key consequence immediately after (§2.2.3, p. 14–15):

“The fact that the category \(S^G\) is equivalent to the category of objects in \(S\) equipped with a G-action has an important and useful consequence. It means that if a construction involving spectra happens to produce something with a G-action, it defines a functor with values in G-spectra.”

The norm functor (HHR, Definition A.52, §A.4)

Using Prop A.19 to identify \(\mathrm{Sp}^H = S^{BH}\) and \(\mathrm{Sp}^G = S^{BG}\), HHR define the norm as follows. Write \(B_{G/H}G\) for the topological G-category with one object and automorphism group G acting on itself, “over” the G-set \(G/H\). The inclusion of the identity coset gives an equivalence \(BH \simeq B_{G/H}G\).

Definition A.52 (HHR). The norm functor \(N_H^G: \mathrm{Sp}^H \to \mathrm{Sp}^G\) is the composite

\[\mathrm{Sp}^H = S^{BH} \xrightarrow{\;i_!\;} S^{B_{G/H}G} \xrightarrow{\;p^\wedge_*\;} S^{BG} = \mathrm{Sp}^G,\]

where \(i_!\) is left Kan extension and \(p^\wedge_*\) is the indexed smash product over the G-set \(G/H\).

Concretely, for \(X \in \mathrm{Sp}^H\) and a choice of coset representatives \(\{g_1, \ldots, g_n\}\) for \(G/H\):

\[N_H^G(X) = \bigwedge_{gH \in G/H} g \cdot X\]

where G acts on this smash product by permuting the factors (via the left action of G on \(G/H\)) and each factor \(g \cdot X\) carries the conjugate H-action. No representation sphere appears anywhere in this definition. The norm is purely an indexed smash product construction, analogous to the Evens norm in group cohomology.

Key property: geometric fixed points (HHR, Proposition 2.55)

The utility of the norm in the Kervaire invariant proof comes from its interaction with geometric fixed points \(\Phi^H\) (§9.5):

Proposition 2.55 (HHR). For \(H \leq G\) and \(X \in \mathrm{Sp}^H\),

\[\Phi^H(N_H^G X) \simeq (\Phi^H X)^{\wedge [G:H]}.\]

In particular, if \(X\) is an \(H\)-equivariant ring spectrum, \(\Phi^H(N_H^G X)\) is the \([G:H]\)-fold smash power of the underlying ring spectrum.

This is the genuine equivariant content: the norm is defined without representation spheres, but Prop 2.55 shows it sees the correct RO(G)-graded information at fixed points. The proof uses that the norm preserves cofibrant objects (Prop B.89) and the formula for geometric fixed points of an indexed smash product.

9.5 Preview: Fixed-Point Functors

The change-of-group functors interact with three distinct fixed-point constructions, which will be treated in full in Fixed-Point Spectra (no note yet):

1. Categorical fixed points \((-)^H: \mathrm{Sp}^G \to \mathrm{Sp}^{W_GH}\): \((E^H)(V) = E(V)^H\). Well-behaved on fibrant objects but not homotopy-invariant on all spectra.

2. Homotopy fixed points \((-)^{hH}: \mathrm{Sp}^G \to \mathrm{Sp}^{W_GH}\): \(E^{hH} = F(EH_+, E)^H\), the \(H\)-equivariant maps from a contractible free \(H\)-space into \(E\). Homotopy-invariant but loses equivariant information.

3. Geometric fixed points \(\Phi^H: \mathrm{Sp}^G \to \mathrm{Sp}^{W_GH}\): defined by smashing with \(\widetilde{E\mathcal{F}[H]}\) (the cofiber of \(E\mathcal{F}[H]_+ \to S^0\) for the family \(\mathcal{F}[H]\) of subgroups not conjugate to \(H\)) and then taking \(H\)-fixed points. Homotopy-invariant and detects equivariant periodicity.

The three are related by the Tate diagram:

where \(E^{tH} = (E^{hH})^{t C_2}\) (the Tate spectrum, see Equivariant Postnikov and Slice for how the slice filtration controls this).

References