Algebraic models for 1-dimensional categories of rational G-spectra

J.P.C.Greenlees

REVIEW 4 major objections 5 minor 3 cited by

Algebraic models for 1-dimensional categories of rational G-spectra

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that for any compact Lie group $G$, the category of rational $G$-spectra whose geometric isotropy lies in a 1-dimensional block $V$ is Quillen equivalent to the derived category of an explicit abelian category…

desk verdict A serious programmatic advance that is probably right, but the proof as written has a real gap at Lemma 9.11 where strong cospan formality is asserted rather than proved. read the letter →

arxiv 2501.11200 v1 pith:MVU7SF7I submitted 2025-01-20 math.AT

J.P.C.Greenlees This is my paper · ORCID

classification math.AT MSC 55P9155N9155U35

keywords rationalequivariantstablehomotopyalgebraicmodelsQuillenequivalence1-dimensionalblockstoralgroupscompactLiesheavesofringssemifreespectra

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to prove that rational $G$-spectra, when their geometric isotropy is restricted to a 1-dimensional block of conjugacy classes, are modelled by a small explicit abelian category. The main theorem states a Quillen equivalence between $G$-spectra$|V$ and $\mathrm{DG}\text{-}\mathcal{A}(G|V)$ whenever $V$ is obtained from a set of finite subgroups by adjoining one conjugacy class with finite Weyl group. This covers all blocks of 1-dimensional groups, semifree spectra for tori, and 1-dimensional blocks of many other groups, unifying earlier results for $SO(2)$ and $O(2)$. A sympathetic reader should care because it turns a homotopy-theoretic category into ordinary algebra, namely sheaves of modules over a cospan of rings, making rational equivariant cohomology computable in these cases.

What carries the argument

The load-bearing object is the standard model $\mathcal{A}(V,R,S,W)$: a category of equivariant sheaves of modules over the cospan of rings $R(K^*)\to S^{-1}O_K\leftarrow O_K$, where $O_K=\prod_{F\in K} R(F)$. An object is a torsion $R(K^*)$-module $V$ (the vertex), an $O_K$-module $N$ (the nub), and an isomorphism $S^{-1}N \cong S^{-1}O_K\otimes V$ enforcing quasicoherence and extendedness, together with compatible actions of the finite component groups. The topological input is the homotopy pullback square $S^0 \to E\langle K^*\rangle$, $DE\langle K\rangle \to E\langle K^*\rangle\wedge DE\langle K\rangle$; the machinery converts this square into the algebraic cospan, and strong intrinsic formality (Lemma 9.11) is what lets homology determine the cospan's homotopy type.

What would settle it

Construct a cospan of commutative ring spectra with the same rational homology as $(S^0)_\flat$—same $R(K^*)$, same $O_K$, same localization $S^{-1}O_K$, same component-group actions—but with a nonzero higher operation such as a Massey product or nontrivial $k$-invariant that changes the derived category of modules. If such a cospan exists, Lemma 9.11 fails and the Quillen equivalence of Theorem 9.1 would not follow from its proof.

Watch

Extended reading notes

Core claim

The central claim is Theorem 9.1: for a compact Lie group $G$, if $V$ is formed from a countable set $K$ of finite subgroups by adjoining a single conjugacy class $K^*$ with finite Weyl group, then there is a Quillen equivalence $G$-spectra$|V \simeq \mathrm{DG}\text{-}\mathcal{A}(G|V)$, where $\mathcal{A}(G|V)$ is equivalent to the standard model $\mathcal{A}(K,R,S,W)$ built from the sheaf of rings, component structure, and coordinate structure described in Section 4. The proof proceeds by exhibiting the sphere spectrum as a homotopy pullback of a cospan, showing the resulting ring spectra are intrinsically formal, and then identifying the cellularization of their module category with the derived category of the abelian model. The paper also shows the abelian category $\mathcal{A}(G|V)$ has injective dimension 1 in the Type 0 and Type 1 cases, and explains how 1-dimensional groups decompose into such blocks.

Load-bearing premise

The argument assumes the cospan of ring spectra replacing the sphere is intrinsically formal in the strong sense of Lemma 9.11: the rational homology rings, the localization, and the component-group actions determine the homotopy type of the ring spectra and their entire module category, and this formality is only sketched in Section 9.C.

Editorial extensions

If this is right

For every 1-dimensional compact Lie group, rational $G$-spectra in each block are Quillen equivalent to the derived category of an explicit abelian category.
Semifree rational $T$-spectra for a torus $T$ admit the same algebraic model, since they sit in 1-dimensional blocks.
The models for $O(2)$ and $Pin(2)$ blocks are recovered uniformly, including the dihedral and quaternion full-subgroup blocks and the toral blocks.
Because $\mathcal{A}(G|V)$ has injective dimension 1, $\mathrm{DG}\text{-}\mathcal{A}(G|V)$ carries the injective model structure with homology isomorphisms as weak equivalences.
The pullback-square presentation of the sphere gives a blueprint for algebraic models in higher-dimensional blocks.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The intrinsic-formality step is likely the first place to test for counterexamples: if a cospan with the same homology but different higher structure exists, the theorem would need a sharper hypothesis.
For $G=SO(3)$ with $K^*=SO(2)$, the paper notes the $S^{\infty V}$ model is unavailable; checking this block explicitly would test whether the proof extends beyond the examples where that model exists.
A computable corollary of the main theorem is that rational equivariant cohomology theories on these blocks are determined by the algebraic data $(R,S,W)$, so Bredon cohomology computations reduce to algebra in these cases.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Let me tell you what I think. This is a serious paper by the right person, and it is not a token contribution: it extends the known circle and O(2) models to all 1-dimensional blocks, semifree toral spectra, and the full blocks of toral groups with simple H1(T;Q) action, with T x C2 and Pin(2) as concrete new cases. The abelian category A(G|V) is set up cleanly, the injective dimension argument in Section 5 is genuine, and the overall strategy—pullback square, strong isotropic formality, cellular skeleton—is coherent and follows the established Greenlees–Shipley architecture. Credit where due: if Theorem 9.1 is correct, this is a major step in the algebraic models program, not a repackaging.

The soft spots are real but concentrated. First, Lemma 2.1, the block decomposition for toral groups, is imported from the unpublished companion [13]. The paper claims it does not logically depend on [13], yet the proof of the decomposition does depend on that lemma. That is a self-containedness problem, and it should be fixed by either including the proof or pointing to a published version.

Second, and more seriously, the stress-test concern about Lemma 9.11 lands. The proof establishes, at best, objectwise formality of the entries. But what the Quillen equivalence needs is formality of the whole cospan, including the map O' -> T' being the derived localization at S. A homology isomorphism H_*(T') ≅ S^{-1}H_*(O') does not force that universal property. The sentence in Section 9.C about 'inverting additional classes already inverted in homology' is exactly where the argument assumes the missing statement. This is sharpest for K0, where S contains infinitely many idempotents and S^{-1}O_K is a localization of an infinite product; that ring is not free graded-commutative, so the Maschke-style splitting used for the generic point cannot be applied there. The proof of Lemma 9.12 also moves quickly, though less dangerously.

Do I think the theorem is false? No. The strategy is plausible and the machinery is well-tested. But the text, as it stands, does not prove strong cospan formality. This is a load-bearing gap, not a cosmetic one.

Who is this for? Specialists in equivariant stable homotopy theory and anyone working on the Greenlees conjecture. A serious referee should see it, but with a clear request: expand Section 9.C, prove the derived localization statement or give a hands-on check in the K0 case, and sort out the status of Lemma 2.1. My recommendation is conditional acceptance with those revisions.

Referee Report

4 major / 5 minor

Summary. The paper constructs algebraic models for rational G-spectra whose geometric isotropy lies in a 1-dimensional block of conjugacy classes, i.e. a space V = K ⊔ {K*} where K is a countable set of finite subgroups and K* has finite Weyl group. The main theorem (Theorem 9.1) states a Quillen equivalence G-spectra|V ≃ DG-A(G|V), where A(G|V) is an explicit abelian category built from a sheaf of rings R, a component structure W, and a multiplicative set S. The proof follows the strategy used for SO(2) and O(2): express the sphere spectrum as a homotopy pullback of an isotropic cospan, pass to modules over the individual ring spectra, use strong isotropic formality to replace them by their homology, and finally identify the cellularization of the resulting module category with DG objects in A(G|V). The paper also proves that the relevant abelian categories have injective dimension 1 in the Type 0 and Type 1 cases, and it works out several explicit examples including tori, O(2), Pin(2), T×C2, and toral blocks related to SU(3).

Significance. If the main theorem is correct, it gives a uniform and calculable algebraic model for all 1-dimensional blocks of rational G-spectra, covering new cases beyond the previously known circle and O(2) results. The paper is well organized, builds on established theorems of Greenlees–Shipley and others, and provides explicit descriptions of the abelian models together with homological dimension estimates. The main gap is the proof of the strong intrinsic formality of the cospan in Section 9.C, which is the load-bearing step that converts spectral input into the explicit algebraic category; as written, this step is only sketched and relies on an unstated universal property of derived localization. Because this gap affects the central Quillen equivalence, the paper is not yet complete, but the overall strategy is plausible and the result is likely to be a significant contribution once the formality proof is supplied.

major comments (4)

[9.C, Lemma 9.11] The proof of intrinsic formality of the cospan is not complete. Lemma 9.11 asserts that a cospan of ring spectra O'_{K*} -> T' <- O' with homology R(K*) -> S^{-1}OK <- OK is intrinsically formal in the strong sense needed for Step 4 of the outline in Section 6.A. The proof only establishes homology isomorphisms entrywise and then says that 'inverting additional classes already inverted in homology induces a weak equivalence'; this is precisely the universal property that must be proved, and a homology isomorphism S^{-1}H_*(O') ≅ H_*(T') does not imply that T' is the derived localization of O' at S. The difficulty is sharpest for the K0 part, where S contains infinitely many idempotents and S^{-1}OK is a localization of an infinite product, a ring that is not free graded-commutative and to which the Maschke/symmetric-algebra argument does not apply. Since Theorem 9.1 depends directly on this cospan-level formality, the proof as written does not establish the Quillen equivalence for the full class stated in the theorem. Please either provide a complete proof, cite a published theorem that covers the cospan case, or state Theorem 9.1 with an explicit intrinsic-formality hypothesis.
[2, Lemma 2.1] Lemma 2.1, the partition of X_G into blocks V^G_H for a toral group G, is imported from the unpublished companion preprint [13]. This lemma is used to justify the paper's claim, stated in the abstract, that the results include all blocks of all groups of dimension 1. Since [13] is not yet available, the paper is not self-contained at this structural point. Please include a proof of Lemma 2.1 in the present paper, or state the main theorem as conditional on [13].
[5.A and Definition 3.7] The definition of the multiplicative set S is inconsistent when KR is nonempty. In Definition 3.7, S_{K*/F} = {0} for F ∈ KR, while in Section 5.A the set S is defined as {(s_F) | s_F ∈ S_{K*/F} and s_F = 1 almost everywhere}. If KR is infinite, no element can satisfy both conditions; if KR is finite and nonempty, the elements of S are forced to be 0 on KR, so inverting S gives the zero ring on that factor, which contradicts the intended product model for height-0 summands described in the introduction. This issue affects Theorem 9.1, which is stated for general V containing KR. Please correct the definition (for example, taking S_{K*/F} = {1} on KR) or restrict the theorem to KR = ∅.
[9, beginning] Theorem 9.1 is stated without any hypothesis that KR = ∅, but the proof begins with the sentence 'Since we know how to deal with 0-dimensional summands, we suppose KR = ∅.' The reduction via Example 3.5(i) is only sketched and is not proved in the paper. It is not obvious that the splitting off of the KR part by idempotents works in full generality for compact Lie groups, particularly when K* is not the whole group G. Please either state Theorem 9.1 under the hypothesis KR = ∅, or give a complete proof of the reduction to that case.

minor comments (5)

[Abstract] There is a typo in the abstract: 'bloc k' should be 'block'.
[6.A] The sentence 'We will first expla The abelian models are Q models gen star' appears to be an unfinished draft and should be removed or completed.
[Lemma 5.3 proof] In the proof of Lemma 5.3, the phrase 'the are non-split extensions' should read 'there are non-split extensions'.
[Example 5.11] In Example 5.11(ii), 'agan' should be 'again'.
[9.C, Lemma 9.11 diagram] The diagram in the proof of Lemma 9.11 is difficult to parse; several labels (e.g., 'T'' = O'_{K*}⊗ T' T' T'' =') appear garbled. Please redraw the diagram and explain which maps are the homology isomorphisms and which are the zigzags proving cospan formality.

Circularity Check

1 steps flagged · score 3.0 of 10

No definitional or fitted-input circularity; the only load-bearing self-citation is Lemma 2.1 imported from the author's companion paper [13], while Lemma 9.11's strong formality is a proof gap rather than a circular reduction.

self citation load bearing [Section 2, Lemma 2.1 (with Section 1.A's note on [13])]
"Lemma 2.1. [13, 2.1] For a toral group G as above, the space XG = Sub(G)/G of conjugacy classes of subgroups of the toral group G is partitioned into pieces,V G H one for each conjugacy class of subgroups H of W."

Section 2 uses this lemma to decompose a 1-dimensional group's subgroup space into blocks of the form covered by Theorem 9.1, so the advertised coverage of 'all blocks of all groups of dimension 1' rests on [13] rather than on a proof in this paper. [13] is the author's own companion preprint, and Section 1.A claims the present paper 'does not logically depend on [13]'. This is load-bearing self-citation for the scope claim, though it does not make the block model itself definitional: the construction of A(G|V) and the cospan argument for Theorem 9.1 are still carried out in this paper.

full rationale

The central derivation is not circular in the forbidden sense: A(G|V) is built from auxiliary data in Section 5, and Theorem 9.1 is proved by expressing G-spectra|V as modules over a pullback cospan, computing the homology of the cospan (Lemmas 9.5 and 9.6), and comparing the cellularized module category to DG-A(G|V) through the cellular skeleton theorem (Lemma 9.12). No parameter is fitted to the target category, and no term of the cospan is defined as the very model the theorem is meant to predict. The main weakness is proof completeness rather than circularity: Lemma 9.11's proof of strong intrinsic formality is only a sketch, and the sentence 'inverting additional classes already inverted in homology induces a weak equivalence' is the load-bearing step that upgrades objectwise formality to the required cospan formality; I treat that as an omitted-proof/correctness risk, not as a circular reduction. The one genuine self-citation issue is Lemma 2.1, which is imported from the same author's companion paper [13] and is load-bearing for the paper's claim to cover all 1-dimensional blocks, so the score is 3 rather than 0-2.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No numerical parameters are fitted to data; the auxiliary data (R,S,W) are determined by the geometry of G. The abelian category A(K,R,S,W) is a new mathematical construction, not a postulated physical entity. The non-canonical decomposition K=K0 union K1 union KR (Remark 3.2) is a choice of presentation, not a fitted parameter.

assumptions (4)

domain assumption For a toral group G, the space XG of conjugacy classes is partitioned into clopen blocks V^G_H indexed by conjugacy classes of subgroups H of the component group W, closed under cotoral specialization (Lemma 2.1).
Used in Section 2.A to reduce the general case to a single 1-dimensional block; stated with citation to the author's preprint [13] without proof in this paper.
domain assumption Upper semicontinuity of normalizers: NG(K) is contained in NG(K*) for almost all subgroups K in K0.
Used in Section 4.A to define the homomorphism R(K*) -> R(F) and the component structure maps for K in K0; cited from [1, 9.8].
standard math The established Quillen equivalence and formality results: fixed-point adjunctions [22], free rational G-spectra [21], modules over diagrams of rings [23], torus models [24], Shipley's theorem [26], and the Cellularization Principle [20].
These published theorems are the backbone of the proof of Theorem 9.1 in Part 2; the paper adapts them to infinite families in Section 8.
standard math Maschke's theorem and the finite-dimensionality of Wd_G(K) (tom Dieck finiteness) are used to split module categories over products of rings and to bound injective dimension.
Invoked in Section 9.C to uniformize over the family K and in Section 5.E to prove the abelian model has injective dimension 1.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Algebraic models for 1-dimensional categories of rational G-spectra." pith.science (2026). https://pith.science/paper/MVU7SF7I

@misc{pith2026250111200,
  author       = {Pith},
  title        = {Pith review of: Algebraic models for 1-dimensional categories of rational G-spectra},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MVU7SF7I}},
  note         = {Machine review of arXiv:2501.11200}
}

read the original abstract

In this paper we give algebraic models for rational G-spectra for a compact Lie group G when the geometric isotropy is restricted to lie in a 1-dimensional block of conjugacy classes. This includes all blocks of all groups of dimension 1, semifree spectra, and 1-dimensional blocks for many other groups G. The results were known previously for G=SO(2) or O(2) due to work of Barnes, Shipley and the author.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Rational G-spectra for rank 2 toral groups of mixed type
math.AT 2025-01 conditional novelty 7.0 of 10

For three rank-2 toral groups of mixed type, rational G-spectra over full subgroups are Quillen equivalent to an explicit abelian category A(G|full).
Rational SU(3)-equivariant cohomology theories
math.AT 2025-02 conditional novelty 6.0 of 10

Rational SU(3)-equivariant spectra are Quillen equivalent to differential graded objects of an explicitly constructed abelian category A(SU(3)).
An algebraic model for rational U(2)-spectra
math.AT 2025-02 conditional novelty 6.0 of 10

The category of rational U(2)-spectra is shown to decompose into seven blocks, each with an explicit algebraic model, yielding a calculable algebraic model for the whole category.

Reference graph

Works this paper leans on

27 extracted references · 15 canonical work pages · cited by 3 Pith papers

[13]

J. P. C. Greenlees. Spaces of subgroups of toral groups. Preprint, 38pp, arXiv:2501.06914 , 2025

arXiv 2025
[1]

Balchin, T

S. Balchin, T. Barthel, and J. P. C. Greenlees. Prismatic decompo sitions and rational G-spectra. Preprint, 59pp, arXiv 2311.18808 , 2023

arXiv 2023
[2]

Scott Balchin and J. P. C. Greenlees. Adelic models for tensor tria ngulated categories. Advances in Mathematics, 375:30pp, 2020

work page 2020
[3]

Scott Balchin and J. P. C. Greenlees. Separated and complete ad elic models for one-dimensional Noe- therian tensor-triangulated categories. J. Pure Appl. Algebra , 226(12):Paper No. 107109, 42, 2022

work page 2022
[4]

Greenlees, and Magdalena K¸ edziorek

David Barnes, J.P.C. Greenlees, and Magdalena K¸ edziorek. An alg ebraic model for rational toral G- spectra. Algebr. Geom. Topol., 19(7):3541–3599, 2019

2019
[5]

J. P. C. Greenlees. A rational splitting theorem for the universa l space for almost free actions. Bull. London Math. Soc. , 28(2):183–189, 1996

work page 1996
[6]

J. P. C. Greenlees. Rational Mackey functors for compact Lie g roups. I. Proc. London Math. Soc. (3) , 76(3):549–578, 1998

work page 1998
[7]

J. P. C. Greenlees. Rational O(2)-equivariant cohomology theo ries. In Stable and unstable homotopy (Toronto, ON, 1996), volume 19 of Fields Inst. Commun. , pages 103–110. Amer. Math. Soc., Providence, RI, 1998

work page 1996

Show all 27 references

[8]

J. P. C. Greenlees. Rational S1-equivariant stable homotopy theory. Mem. Amer. Math. Soc. , 138(661):xii+289, 1999

1999
[9]

J. P. C. Greenlees. Triangulated categories of rational equivar iant cohomology theories. Oberwolfach Reports, pages 480–488, 2006. (cit. on p. 2) , 2006

2006
[10]

J. P. C. Greenlees. Rational equivariant cohomology theories w ith toral support. Algebr. Geom. Topol., 16(4):1953–2019, 2016

1953
[11]

J. P. C. Greenlees. Borel cohomology and the relative gorenst ein condition for classifying spaces of compact lie groups. Preprint, page 10, 2018

2018
[12]

J. P. C. Greenlees. The Balmer spectrum of rational equivarian t cohomology theories. J. Pure Appl. Algebra, 223(7):2845–2871, 2019

2019
[14]

J. P. C. Greenlees. Rational G-spectra for small toral groups. Preprint, 31pp, 2025

2025
[15]

J. P. C. Greenlees. An algebraic model for rational U (2)-spectra. In preparation, 15pp , 2025

2025
[16]

J. P. C. Greenlees. An algebraic model for rational SU (3)-spectra. In preparation, 11pp , 2025

2025
[17]

J. P. C. Greenlees. Rational G-spectra for components with ﬁnite Weyl groups. In preparation, 4pp , 2025. 27

2025
[18]

J. P. C. Greenlees. An abelian model for rational G-spectra for a compact Lie group G. In preparation, 27pp
[19]

J. P. C. Greenlees. An algebraic model for rational G-spectra for ﬁnite central extensions of a torus. In preparation, 18pp
[20]

J. P. C. Greenlees and B. Shipley. The cellularization principle for Q uillen adjunctions. Homology Ho- motopy Appl. , 15(2):173–184, 2013

2013
[21]

J. P. C. Greenlees and B. Shipley. An algebraic model for free ra tional G-spectra. Bull. Lond. Math. Soc., 46(1):133–142, 2014

2014
[22]

J. P. C. Greenlees and B. Shipley. Fixed point adjunctions for eq uivariant module spectra. Algebr. Geom. Topol., 14(3):1779–1799, 2014

2014
[23]

J. P. C. Greenlees and B. Shipley. Homotopy theory of modules o ver diagrams of rings. Proc. Amer. Math. Soc. Ser. B , 1:89–104, 2014

2014
[24]

J. P. C. Greenlees and B. Shipley. An algebraic model for rationa l torus-equivariant spectra. J. Topol., 11(3):666–719, 2018

2018
[25]

The homotopy theory of complet e modules

Luca Pol and Jordan Williamson. The homotopy theory of complet e modules. J. Algebra , 594:74–100, 2022

2022
[26]

HZ-algebra spectra are diﬀerential graded alg ebras

Brooke Shipley. HZ-algebra spectra are diﬀerential graded alg ebras. American Journal of Mathematics , 129(2):351–379, 2007

2007
[27]

Transformation groups and representation theory , volume 766 of Lecture Notes in Mathematics

Tammo tom Dieck. Transformation groups and representation theory , volume 766 of Lecture Notes in Mathematics. Springer, Berlin, 1979. Mathematics Institute, Zeeman Building, Coventry CV4, 7AL , UK Email address : john.greenlees@warwick.ac.uk 28

1979

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [13]

J. P. C. Greenlees. Spaces of subgroups of toral groups. Preprint, 38pp, arXiv:2501.06914 , 2025

arXiv 2025

[2] [1]

Balchin, T

S. Balchin, T. Barthel, and J. P. C. Greenlees. Prismatic decompo sitions and rational G-spectra. Preprint, 59pp, arXiv 2311.18808 , 2023

arXiv 2023

[3] [2]

Scott Balchin and J. P. C. Greenlees. Adelic models for tensor tria ngulated categories. Advances in Mathematics, 375:30pp, 2020

work page 2020

[4] [3]

Scott Balchin and J. P. C. Greenlees. Separated and complete ad elic models for one-dimensional Noe- therian tensor-triangulated categories. J. Pure Appl. Algebra , 226(12):Paper No. 107109, 42, 2022

work page 2022

[5] [4]

Greenlees, and Magdalena K¸ edziorek

David Barnes, J.P.C. Greenlees, and Magdalena K¸ edziorek. An alg ebraic model for rational toral G- spectra. Algebr. Geom. Topol., 19(7):3541–3599, 2019

2019

[6] [5]

J. P. C. Greenlees. A rational splitting theorem for the universa l space for almost free actions. Bull. London Math. Soc. , 28(2):183–189, 1996

work page 1996

[7] [6]

J. P. C. Greenlees. Rational Mackey functors for compact Lie g roups. I. Proc. London Math. Soc. (3) , 76(3):549–578, 1998

work page 1998

[8] [7]

J. P. C. Greenlees. Rational O(2)-equivariant cohomology theo ries. In Stable and unstable homotopy (Toronto, ON, 1996), volume 19 of Fields Inst. Commun. , pages 103–110. Amer. Math. Soc., Providence, RI, 1998

work page 1996