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The Lubin-Tate Theory of Configuration Spaces: I

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Pith's one-line read A convergent spectral sequence computes Morava E-theory of labeled configuration spaces, with its E2 page identified as the homology of a Hecke–Chevalley–Eilenberg complex.

desk verdict Genuinely new computational machinery for Morava E-homology of configuration spaces, but the main theorem rests on an unproved completion-to-isomorphism upgrade from the first author's thesis. read the letter →

arxiv 1908.11321 v4 pith:JD3SFCO3 submitted 2019-08-29 math.AT math.GT

Lukas Brantner , Jeremy Hahn , Ben Knudsen This is my paper · ORCID
classification math.ATmath.GT MSC 55R8055P4855N2055T99
keywords MoravaE-theoryconfigurationspacesHeckeLiealgebrasspectralsequencesLubin-TatetheoryiteratedloopDyer-Lashofoperationssurfaceconfigurations
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

This paper introduces a spectral sequence that computes the Morava E-theory of unordered configuration spaces at odd primes and any chromatic height. The E2 page is identified with the homology of a Chevalley–Eilenberg-style complex built from Hecke Lie algebras, algebraic objects that package the additive operations acting on completed E-homology. From this general tool the authors derive explicit formulas for the E-theory of the weight p summands of iterated loop spaces of spheres and of configurations of p points on punctured surfaces. These computations yield Morava K-theory groups relevant to a longstanding conjecture on loop spaces and, in the limit as height tends to infinity, new results on ordinary mod p homology of configuration spaces of surfaces. The paper's central claim is that this spectral sequence exists, has the stated algebraic E2 page, and is powerful enough to make these computations routine.

What carries the argument

The load-bearing object is the Hecke Lie algebra $g(M;X) = E^{\wedge}_*(\mathrm{FreeLie}(\Sigma^{n-1}X)^{M_+})$, together with its Hecke–Chevalley–Eilenberg complex $CE_{Hu}$. A Hecke Lie algebra is a Lie algebra in $E_*$-modules equipped with an action of the ring of additive power operations, and $CE_{Hu}$ is the complex obtained by killing those operations in a derived fashion and then forming the derived abelianisation. The argument converts a stable equivalence between configuration spaces and the bar construction on a free spectral Lie algebra into an algebraic spectral sequence, using a theorem from the first author's thesis that identifies the completed E-homology of a free K(h)-local spectral Lie algebra with the free Hecke Lie algebra on its homotopy groups.

What would settle it

Compute the completed E-homology of $B_3(\mathbb{R}^3)$ at height 2 and p=3 by an independent method, such as a Gysin sequence for the symmetric group action or a direct Atiyah–Hirzebruch computation, and compare with the paper's prediction $E_* \oplus \Sigma^{-1}E_*/3$; a different torsion summand would falsify the spectral sequence's E2-page identification.

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Extended reading notes

Core claim

The paper's central claim is that for a framed n-manifold M and a spectrum X, whenever the completed E-homology Hecke Lie algebra $g(M;X)$ is finite and free in each weight, there is a convergent weighted spectral sequence $E^2_{s,t} \cong H_{s+1}(CE_{Hu}(g(M;X)))_{t-1}$ converging to $\bigoplus_{k\ge 0} E^{\wedge}_{s+t}(B_k(M;X))$, where $B_k(M;X)$ is the labeled unordered configuration space. Here $CE_{Hu}$ is a Hecke analogue of the Chevalley–Eilenberg complex: it first kills the additive operations in a derived fashion and then takes the derived abelianisation of the resulting Lie algebra. The authors apply this spectral sequence to compute the completed E-homology of the pth Snaith summand of $\Omega^n S^{n+k}$ for all n, k, and all heights, and the E-cohomology of the space of p unordered points in a punctured orientable surface. They read off the corresponding Morava K-theory groups and, by letting the height grow, prove that the integral homology of $B_p(S_{g,1})$ has no p-power torsion.

Load-bearing premise

The whole computation rests on an unproved theorem quoted from the first author's thesis: the completed E-homology of a free spectral Lie algebra is exactly the free Hecke Lie algebra on its homotopy groups, and the paper does not reprove it.

Editorial extensions

If this is right

  • The E-theory of the pth weight summand of $\Omega^n S^{n+k}$ is now known explicitly for every n, k, and height h at odd primes, giving a concrete description of the weight p operations on $E_n$-algebras.
  • For punctured orientable surfaces, the E-cohomology of $B_p(S_{g,1})$ is a direct sum of free $E_*$-modules with explicit ranks, so the E-theory of these configuration spaces is torsion-free.
  • The same surface computation implies that the integral homology of $B_p(S_{g,1})$ has no p-power torsion, so its $\mathbb{F}_p$ Betti numbers coincide with its rational Betti numbers.
  • At height 1 the spectral sequence recovers known p-adic K-theory computations, and as h tends to infinity it yields new $\mathbb{F}_p$-homology computations for configurations of p points on surfaces.
  • The Morava K-theory groups read off from the spectral sequence show that the size of the weight p part of $K(h)_*(\Omega^n S^{n+1})$ depends on height for n at least 4, which runs against the pattern suggested by earlier conjectures.

Reading between the lines

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

  • The same spectral sequence, with a more careful treatment of the coherent coalgebra structure, should extend to weights $p^i$ and could settle the full conjecture at all weights.
  • The torsion summand of the form $E_*(B\Sigma_p)/(tr)$ suggests that weight p E-theory of configuration spaces is governed by the modular representation theory of the symmetric group, a connection the paper leaves mostly implicit.
  • A direct testable extension is to apply the method to non-framed manifolds after incorporating tangential structure groups, following the authors' own stated future direction, and to compare the results against known rational Betti numbers.
  • For explicit height 2 examples, the formulas give concrete answers such as the p=3 computation $E^{\wedge}_*(B_3(\mathbb{R}^{11})) \cong E_* \oplus \Sigma^{-1}E_*[\alpha]/(\alpha^4 - 6\alpha^2 + (h-9)\alpha - 3, \alpha^5)$, which could be checked independently through a direct analysis of Dyer–Lashof operations.
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Editorial analysis

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

Referee Report

1 major / 4 minor

Summary. This paper develops a new method for computing the completed Morava E-homology of unordered configuration spaces of framed manifolds. The main structural result, Theorem 5.5, is a weighted spectral sequence converging to ⊕_k E^∧_{s+t}(B_k(M;X)), whose E2-page is identified with the homology of a Hecke–Chevalley–Eilenberg complex for an unshifted Hecke Lie algebra g(M;X). The construction combines Knudsen's stable splitting of configuration spaces (Theorem 5.1) with Brantner's description of the E-homology of free K(h)-local spectral Lie algebras in terms of Hecke Lie algebras (Theorem 4.20). Using this tool, the paper computes the E-theory of the weight-p Snaith summands of iterated loop spaces of spheres at arbitrary height (Theorem 6.10), derives the corresponding Morava K-theory groups (Theorem 6.19), and computes the E-theory and F_p-homology of configuration spaces of p points on punctured surfaces (Theorems 7.1 and 1.10). The paper also proves several foundational algebraic results, including the Chevalley–Eilenberg computation for flat Lie algebras (Theorem 3.13) and the existence of transferred model structures for Hecke Lie algebras.

Significance. If the identification of the E2-page is fully justified, the paper supplies a general and explicit computational tool for Morava E-theory of labeled configuration spaces, with applications to power operations on E_n-algebras and to longstanding conjectures of Ravenel. The paper is carefully written and contains complete proofs of the algebraic foundations in Sections 2–4, including the Chevalley–Eilenberg theorem for flat Lie algebras and the construction of the Hecke–Chevalley–Eilenberg complex. The derived computations are explicit and falsifiable, and they match independent results at height 1 (Langsetmo) and classical F_p results (Cohen). The main gap is the unproved upgrade from a completion isomorphism to an honest isomorphism in Corollary 4.21, which is the load-bearing bridge to the spectral sequence.

major comments (1)
  1. [Section 4.5, Corollary 4.21; used in Section 5.2, proof of Theorem 5.5] The E2-page identification of the Hecke spectral sequence rests on the assertion that the canonical map FreeLie_Hu(π_* M) → π_* FreeLE(L_K(h) M) is an isomorphism for M in the category of pointwise finite free positive-weight E-module spectra, whereas Theorem 4.20(2) only provides this isomorphism after completion. Corollary 4.21 is stated without proof, with only a reference to [Bra17, §4.4.2]. This upgrade is load-bearing: Theorem 5.5 applies it levelwise to replace the homotopy of the topological bar construction by the algebraic bar construction Bar(id, L_Hu, g(M;X)) and then by CE_Hu(g). The height-1 comparison in §6.1 gives indirect evidence but does not test cases in which the completed and uncompleted answers differ. I recommend adding a proof of Corollary 4.21, or at least a detailed account of the finiteness and naturality of the cotensor construction that yields the honest isomorphism.
minor comments (4)
  1. [Section 5.2, proof of Theorem 5.5] The expression "Bar•(LHu, ,LHu, g(M;X))" contains an extra comma, and the reference to "Proposition 4.21" should be "Corollary 4.21."
  2. [Section 6.1, proof of Theorem 6.1] In the discussion of the map E2_0,*(p,1,k) → E2_0,*(p,n,k) for n odd and k even, the map is written as multiplication by p^{n/2}, but n is odd in that case; the exponent should be floor(n/2) or the description of the induced map on the torsion subgroups should be clarified.
  3. [Section 7.1, Theorem 7.1(1)] The formula for E∗(Bp( ˙T ))) has an extra closing parenthesis; it should be E∗(Bp( ˙T )).
  4. [Remark 1.4] The expression "α 5" should presumably be "α^5"; the exponent is missing a caret.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the E2-page identification rests on cited prior theorems with independent content, and the derived computations are not fed back into the definitions.

full rationale

The derivation chain is: Knudsen's stable splitting (Theorem 5.1) expresses labeled configuration spaces as a bar construction on a free spectral Lie algebra; Brantner's Theorem 4.20 identifies the homotopy of free K(h)-local spectral Lie algebras with free Hecke Lie algebras up to completion, upgraded in Corollary 4.21 for finite-free positive-weight modules; the algebraic Chevalley–Eilenberg complex for Hecke Lie algebras is developed and proven self-contained in Sections 3–4; the spectral sequence of Theorem 5.5 assembles these inputs. None of these steps defines the target in terms of itself. The Hecke Lie algebra g(M;X) is a topological invariant (completed E-homology of a spectral Lie algebra), not a fitted parameter; the computed E2-pages are derived from the algebraic complex, and the outputs (Theorems 6.10 and 7.1) are not used to define the inputs. The heavy self-citations (Brantner's thesis for Theorem 4.20, Knudsen's paper for Theorem 5.1) are load-bearing, but they are parameter-free theorems with stated assumptions that do not include the configuration-space results, and they are partially calibrated externally by the height-1 comparison with Langsetmo and the F_p-homology match with Cohen. The main soft spot is the completion-to-isomorphism upgrade in Corollary 4.21, which is asserted rather than fully proved in this paper; that is a correctness or completeness concern, not a circularity, because the upgrade is not obtained by assuming the conclusion of Theorem 5.5 or by renaming an input. No fitted parameter is relabeled as a prediction, and no uniqueness theorem is imported to force a choice. The paper's internal algebraic development in Sections 2–4 is self-contained, so the central claim has independent content.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

All inputs are standard objects in chromatic homotopy theory; the nonstandard inputs are the Hecke Lie algebra theory from Brantner's thesis and the stable splitting from Knudsen's work, both cited rather than proved here.

assumptions (5)
  • standard math The category of weighted graded modules over E_* admits transferred model structures on simplicial Lie algebras and Hecke Lie algebras.
    Homotopical algebra background; the paper gives proof sketches and citations to [JN14].
  • domain assumption Theorem 4.20 (Brantner): homotopy groups of free K(h)-local spectral Lie algebras are computed by free Hecke Lie algebras after completion.
    Quoted from [Bra17], not proved here; it is the load-bearing bridge from spectra to algebra.
  • domain assumption Theorem 5.1 (Knudsen): stable equivalence between labeled configuration spaces and the bar construction on free spectral Lie algebras.
    Quoted from [Knu18], not proved here; it underlies the spectral sequence.
  • domain assumption Manifolds considered (R^n and punctured orientable surfaces) are framed and stably split.
    Needed to apply Lemma 5.11 and Proposition 5.9; true for these manifolds.
  • domain assumption p is an odd prime and 2 is invertible in E_*, so divided powers and Lie algebra axioms behave classically.
    The entire paper fixes p > 2; the p = 2 case is explicitly deferred.
invented entities (2)
  • Unshifted Hecke Lie algebra g^u
    purpose: Encodes operations on E-homology of spectral Lie algebras; central object of the E2-page.
    Introduced in [Bra17]; paper uses an unshifted variant. No external falsifiable prediction outside the mathematical framework.
  • Hecke Chevalley-Eilenberg complex CE_Hu
    purpose: Computes Hecke Lie algebra homology; provides the explicit E2-page.
    A new algebraic complex defined in Section 4.4; its output is checked against known cases.

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Pith. "Pith review of The Lubin-Tate Theory of Configuration Spaces: I." pith.science (2026). https://pith.science/paper/JD3SFCO3

@misc{pith2026190811321,
  author       = {Pith},
  title        = {Pith review of: The Lubin-Tate Theory of Configuration Spaces: I},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JD3SFCO3}},
  note         = {Machine review of arXiv:1908.11321}
}
abstract

We construct a spectral sequence converging to the Morava $E$-theory of unordered configuration spaces and identify its E$^2$-page as the homology of a Chevalley-Eilenberg-like complex for Hecke Lie algebras. Based on this, we compute the $E$-theory of the weight $p$ summands of iterated loop spaces of spheres (parametrising the weight $p$ operations on $\mathbb{E}_n$-algebras), as well as the $E$-theory of the configuration spaces of $p$ points on a punctured surface. We read off the corresponding Morava $K$-theory groups, which appear in a conjecture by Ravenel. Finally, we compute the $\mathbb{F}_p$-homology of the space of unordered configurations of $p$ particles on a punctured surface.

Figures

Figures reproduced from arXiv: 1908.11321 by the authors.

Figure 1
Figure 1. The Hecke Lie algebra homology HLieHu (g(R n, Sk ))(p). Hence, the E2-term E2 s,t = HLieHu s+1 (g(R n, Sk ))t−1 of the spectral sequence at weight p is given by: E 2 0,∗ (p) = HLieHu 1 (g(R n , Sk ))(p))∗−1 =    Σ k−1E−∗(BΣp)/(tr, ed n 2 e ) for n even, k even Σ k−1E−∗(BΣp)/(tr, eb n 2 c ) for n even, k odd Σ k−1E−∗(BΣp)/(tr, ed n 2 e ) for n odd, k even Σ k−1E−∗(BΣp)/(tr, eb n 2 c ) for n odd, k odd E 2 p−… view at source ↗

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Works this paper leans on

80 extracted references · 74 canonical work pages

  1. [1]

    Antol \' n Camarena

    O. Antol \' n Camarena. The mod 2 homology of free spectral L ie algebras . PhD thesis, Harvard University, 2015

    work page 2015

  2. [2]

    Ayala and J

    D. Ayala and J. Francis. Factorization homology of topological manifolds. J. Topol. , 8:1045--1084, 2015

    work page 2015

  3. [3]

    M. Barr. Cartan--Eilenberg cohomology and triples. J. Pure Appl. Algebra , 112:219--238, 1996

    work page 1996

  4. [4]

    Rational cohomology of configuration spaces of surfaces

    Carl-Friedrich B \"o digheimer and Fred Cohen. Rational cohomology of configuration spaces of surfaces. In Algebraic Topology and Transformation Groups , volume 1361 of Lecture Notes in Math. , pages 7--13. Springer, 1988

    work page 1988

  5. [5]

    B\" o digheimer, F

    C.-F. B\" o digheimer, F. R. Cohen, and R. J. Milgram. Truncated symmetric products and configuration spaces. Math. Z. , 214(2):179--216, 1993

    work page 1993

  6. [6]

    B \"o digheimer, F

    C.-F. B \"o digheimer, F. R. Cohen, and L. R. Taylor. On the homology of configuration spaces. Topology , 28:111--123, 1989

    work page 1989

  7. [7]

    Beilinson and V

    A. Beilinson and V. Drinfeld. Chiral Algebras . American Mathematical Society, 2004

    work page 2004

  8. [8]

    A. K. Bousfield and D. M. Kan. Homotopy limits, completions and localizations . Lecture Notes in Mathematics, Vol. 304. Springer-Verlag, Berlin-New York, 1972

    work page 1972

Show all 80 references
  1. [9]

    R. R. Bruner, J. P. May, J. E. McClure, and M. Steinberger. H ring spectra and their applications , volume 1176 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, 1986

    1986

  2. [10]

    B \" o digheimer

    C.-F. B \" o digheimer. Stable splittings of mapping spaces. In Algebraic Topology , volume 1286 of Lecture Notes in Math. Springer, 1987

    1987

  3. [11]

    A. K. Bousfield. Localization and periodicity in unstable homotopy theory. J. Amer. Math. Soc. , 7(4):831--873, 1994

    1994

  4. [12]

    Behrens and C

    M. Behrens and C. Rezk. Spectral algebra models of unstable v_n -periodic homotopy theory. arXiv preprint arXiv:1703.02186 , 2017

    2017 arXiv

  5. [13]

    D. L. B. Brantner. The Lubin-Tate Theory of Spectral Lie Algebras . PhD thesis, Harvard University, 2017. Edited Version available at https://people.maths.ox.ac.uk/brantner/brantnerthesis.pdf https://people.maths.ox.ac.uk/brantner/brantnerthesis.pdf

    2017

  6. [14]

    W. Browder. Homology operations and loop spaces. Illinois journal of mathematics , 4(3):347--357, 1960

    1960

  7. [15]

    Chevalley and S

    C. Chevalley and S. Eilenberg. Cohomology theory of Lie groups and Lie algebras . Trans. Amer. Math. Soc. , 1948

    1948

  8. [16]

    M. Ching. Bar constructions for topological operads and the G oodwillie derivatives of the identity. Geom. Topol. , 9:833--933, 2005

    2005

  9. [17]

    F. R. Cohen, T. J. Lada, and J. P. May. Homology of Iterated Loop Spaces . Number 533 in Lecture Notes in Math. Springer, 1976

    1976

  10. [18]

    F. R. Cohen, J. P. May, and L. R. Taylor. Splitting of certain spaces CX . Math. Proc. Camb. Phil. Soc. , 84, 1978

    1978

  11. [19]

    G. C. Drummond-Cole and B. Knudsen. Betti numbers of configuration spaces of surfaces. J. London Math. Soc. , 2017

    2017

  12. [20]

    Dyer and R

    E. Dyer and R. K. Lashof. Homology of iterated loop spaces. Amer. J. Math. , 84:35--88, 1962

    1962

  13. [21]

    Non-additive functors, their derived functors, and the suspension homomorphism

    Albrecht Dold and Dieter Puppe. Non-additive functors, their derived functors, and the suspension homomorphism. Proc. Nat. Acad. Sci. U.S.A. , 44:1065--1068, 1958

    1958

  14. [22]

    P. I. Etingof, I. Frenkel, and A. A. Kirillov. Lectures on representation theory and Knizhnik--Zamolodchikov equations . American Mathematical Soc., 1998

    1998

  15. [23]

    Fossum and H.-B

    R. Fossum and H.-B. Foxby. The category of graded modules. Math. Scand. , 1974

    1974

  16. [24]

    Francis and D

    J. Francis and D. Gaitsgory. Chiral K oszul duality. Selecta Math. , 18(1):27--87, 2012

    2012

  17. [25]

    Rational B etti numbers of configuration spaces

    Yves F\'elix and Jean-Claude Thomas. Rational B etti numbers of configuration spaces. Topology Appl. , 102:139--149, 2000

    2000

  18. [26]

    P. G. Goerss and M. J. Hopkins. Moduli spaces of commutative ring spectra. In Structured ring spectra , volume 315 of London Math. Soc. Lecture Note Ser. , pages 151--200. Cambridge Univ. Press, Cambridge, 2004

    2004

  19. [27]

    P. G. Goerss and J. F. Jardine. Simplicial Homotopy Theory . Birkh \" a user, 2009

    2009

  20. [28]

    S. Glasman. Day convolution for -categories. Mathematical Research Letters , 23(5):1369--1385, 2016

    2016

  21. [29]

    Paul G. Goerss. Hopf rings, D ieudonn\' e modules, and E_* ^2S^3 . In Homotopy invariant algebraic structures ( B altimore, MD , 1998) , volume 239 of Contemp. Math. , pages 115--174. Amer. Math. Soc., Providence, RI, 1999

    1998

  22. [30]

    Universal homotopy associative, homotopy commutative H -spaces and the EHP spectral sequence

    Jelena Grbi\' c . Universal homotopy associative, homotopy commutative H -spaces and the EHP spectral sequence. Math. Proc. Cambridge Philos. Soc. , 140(3):377--400, 2006

    2006

  23. [31]

    Model categories and simplicial methods

    Paul Goerss and Kristen Schemmerhorn. Model categories and simplicial methods. In Interactions between homotopy theory and algebra , volume 436 of Contemp. Math. , pages 3--49. Amer. Math. Soc., Providence, RI, 2007

    2007

  24. [32]

    Lie algebras and v\_n -periodic spaces

    Gijs Heuts. Lie algebras and v\_n -periodic spaces. arXiv preprint arXiv:1803.06325 , 2018

    2018 arXiv

  25. [33]

    P. J. Higgins. Baer invariants and the B irkhoff- W itt theorem. Journal of Algebra , 11:469--482, 1969

    1969

  26. [34]

    Hopkins, N

    M. Hopkins, N. Kuhn, and D. C. Ravenel. Generalized group characters and complex oriented cohomology theories. Journal of the American Mathematical Society , 13(3):553--594, 2000

    2000

  27. [35]

    M. Hopkins. K(1) -local E_ -ring spectra. In Topological modular forms , volume 201 of Math. Surveys Monogr. , pages 287--302. Amer. Math. Soc., Providence, RI, 2014

    2014

  28. [36]

    On the homology spectral sequence for topological hochschild homology

    Thomas Hunter. On the homology spectral sequence for topological hochschild homology. Transactions of the American Mathematical Society , 348(10):3941--3953, 1996

    1996

  29. [37]

    Johnson and J

    N. Johnson and J. Noel. Lifting homotopy t-algebra maps to strict maps. Advances in Mathematics , 264:593--645, 2014

    2014

  30. [38]

    Kudo and S

    T. Kudo and S. Araki. Topology of H_n -spaces and H -squaring operations. Mem. Fac. Sci. Ky\=usy\=u Univ. Ser. A. , 10:85--120, 1956

    1956

  31. [39]

    J. J. Kjaer. On the odd primary homology of free algebras over the spectral Li e operad. arXiv preprint arXiv:1608.06605 , 2016

    2016 arXiv

  32. [40]

    B. Knudsen. Betti numbers and stability for configuration spaces via factorization homology. Alg. Geom. Topol. , 17(5):3137--3187, 2017

    2017

  33. [41]

    B. Knudsen. Higher enveloping algebras. Geom. Topol. , 22(7):4013--4066, 2018

    2018

  34. [42]

    Langsetmo

    L. Langsetmo. The K -theory localization of loops on an odd sphere and applications. Topology , 32(3):577--585, 1993

    1993

  35. [43]

    Langsetmo

    L. Langsetmo. Further structure in K(1)_* ^kS^ 2n+1 . In Algebraic topology: new trends in localization and periodicity ( S ant F eliu de G u\' xols, 1994) , volume 136 of Progr. Math. , pages 235--240. Birkh \" a user, Basel, 1996

    1994

  36. [44]

    Further structure in K(1)_* ^kS^ 2n+1

    Lisa Langsetmo. Further structure in K(1)_* ^kS^ 2n+1 . In Algebraic topology: new trends in localization and periodicity ( S ant F eliu de G u\' xols, 1994) , volume 136 of Progr. Math. , pages 235--240. Birkh\" a user, Basel, 1996

    1994

  37. [45]

    Ass de modules plats

    Daniel Lazard. Ass de modules plats. C. R. Acad. Sci. Paris S\' e r. A-B , 269:A65--A67, 1969

    1969

  38. [46]

    Longoni and P

    R. Longoni and P. Salvatore. Configuration spaces are not homotopy invariant. Topology , 44:375--380, 2005

    2005

  39. [47]

    J. Lurie. Higher algebra. Preprint, http://www.math.harvard.edu/ lurie/papers/higheralgebra.pdf, 2003

    2003

  40. [48]

    S. Marsh. The M orava E -theories of finite general linear groups. arXiv preprint arXiv:1001.1949 , 2010

    1949 arXiv

  41. [49]

    The cohomology of restricted lie algebras and of hopf algebras

    J Peter May. The cohomology of restricted lie algebras and of hopf algebras. Journal of Algebra , 3(2):123--146, 1966

    1966

  42. [50]

    J. G. Mayorquin. The mod 2 K -homology of ^3S^3X . Canad. J. Math. , 40(1):142--196, 1988

    1988

  43. [51]

    McCleary

    J. McCleary. A User's Guide to Spectral Sequences , volume 58 of Cambridge Studies in Advances Mathematics . Cambridge University Press, 2000

    2000

  44. [52]

    D. McDuff. Configuration spaces of positive and negative particles. Topology , 14:91--107, 1975

    1975

  45. [53]

    James Milgram and Peter L\" o ffler

    R. James Milgram and Peter L\" o ffler. The structure of deleted symmetric products. In Braids ( S anta C ruz, CA , 1986) , volume 78 of Contemp. Math. , pages 415--424. Amer. Math. Soc., Providence, RI, 1988

    1986

  46. [54]

    Mahowald and Douglas C

    Mark E. Mahowald and Douglas C. Ravenel. Toward a global understanding of the homotopy groups of spheres. In The L efschetz centennial conference, P art II ( M exico C ity, 1984) , volume 58 of Contemp. Math. , pages 57--74. Amer. Math. Soc., Providence, RI, 1987

    1984

  47. [55]

    On the K -theory of the K ahn- P riddy map

    Haynes Miller and Victor Snaith. On the K -theory of the K ahn- P riddy map. J. London Math. Soc. (2) , 20(2):339--342, 1979

    1979

  48. [56]

    K -theory and unstable homotopy groups

    Mark Mahowald and Robert Thompson. K -theory and unstable homotopy groups. In Algebraic topology ( E vanston, IL , 1988) , volume 96 of Contemp. Math. , pages 273--279. Amer. Math. Soc., Providence, RI, 1989

    1988

  49. [57]

    Thompson

    Mark Mahowald and Robert D. Thompson. The K -theory localization of an unstable sphere. Topology , 31(1):133--141, 1992

    1992

  50. [58]

    Neisendorfer

    J. Neisendorfer. Algebraic Methods in Unstable Homotopy Theory . Cambridge University Press, 2010

    2010

  51. [59]

    Stewart B. Priddy. Koszul resolutions. Trans. Amer. Math. Soc. , 152:39--60, 1970

    1970

  52. [60]

    D. Quillen. On the (co-) homology of commutative rings. In Applications of C ategorical A lgebra ( P roc. S ympos. P ure M ath., V ol. XVII , N ew Y ork, 1968) , pages 65--87. Amer. Math. Soc., Providence, R.I., 1970

    1968

  53. [61]

    D. C. Ravenel. The homology and M orava K -theory of ^2 SU (n) . Forum Math. , 5(1):1--21, 1993

    1993

  54. [62]

    D. C. Ravenel. What we still don't know about loop spaces of spheres. Contemporary Mathematics , 220:275--292, 1998

    1998

  55. [63]

    C. Rezk. Notes on the H opkins- M iller theorem. In Homotopy theory via algebraic geometry and group representations ( E vanston, IL , 1997) , volume 220 of Contemp. Math. , pages 313--366. Amer. Math. Soc., Providence, RI, 1998

    1997

  56. [64]

    The units of a ring spectrum and a logarithmic cohomology operation

    Charles Rezk. The units of a ring spectrum and a logarithmic cohomology operation. J. Amer. Math. Soc. , 19(4):969--1014, 2006

    2006

  57. [65]

    C. Rezk. The congruence criterion for power operations in M orava E -theory. Homology Homotopy Appl. , 11(2):327--379, 2009

    2009

  58. [66]

    C. Rezk. Rings of power operations for M orava E -theories are K oszul. arXiv preprint arXiv:1204.4831 , 2012

    2012 arXiv

  59. [67]

    Lois polynomes et lois formelles en th \'e orie des modules

    Norbert Roby. Lois polynomes et lois formelles en th \'e orie des modules. In Annales scientifiques de l' \'E cole Normale Sup \'e rieure , volume 80, pages 213--348, 1963

    1963

  60. [68]

    Configuration operads, minimal models and rational curves

    Paolo Salvatore. Configuration operads, minimal models and rational curves. PhD thesis, University of Oxford, 1998

    1998

  61. [69]

    Salvatore

    P. Salvatore. Configuration spaces with summable labels. In Cohomological Methods in Homotopy Theory , volume 196 of Progr. Math. Birkh \" a user, 2001

    2001

  62. [70]

    Schiessl

    C. Schiessl. Integral cohomology of configuration spaces of the sphere. arXiv:801.04273

  63. [71]

    J.-P. Serre. Homologie singuli \` e re des espaces fibr \' e s. Ann. of Math. (2) , 54(3):425--505, 1951

    1951

  64. [72]

    Groupes d'homotopie et classes de groupes ab \'e liens

    Jean-Pierre Serre. Groupes d'homotopie et classes de groupes ab \'e liens. Ann. of Math , 58(2):258--294, 1953

    1953

  65. [73]

    V. P. Snaith. A stable decomposition of n S n X . J. London Math. Soc. (2) , 7:577--583, 1974

    1974

  66. [74]

    N. P. Strickland. Morava E -theory of symmetric groups. Topology , 37(4):757--779, 1998

    1998

  67. [75]

    A dual R othenberg- S teenrod spectral sequence

    Dai Tamaki. A dual R othenberg- S teenrod spectral sequence. Topology , 33(4):631--662, 1994

    1994

  68. [76]

    The fiber of iterated F reudenthal suspension and M orava K -theory of ^kS^ 2l+1

    Dai Tamaki. The fiber of iterated F reudenthal suspension and M orava K -theory of ^kS^ 2l+1 . In Recent progress in homotopy theory ( B altimore, MD , 2000) , volume 293 of Contemp. Math. , pages 299--329. Amer. Math. Soc., Providence, RI, 2002

    2000

  69. [77]

    Configuration spaces of algebraic varieties

    Burt Totaro. Configuration spaces of algebraic varieties. Topology , 35(4):1057--1067, 1996

    1996

  70. [78]

    Lambda-rings, binomial domains, and vector bundles over C P( )

    Clarence Wilkerson. Lambda-rings, binomial domains, and vector bundles over C P( ) . Comm. Algebra , 10(3):311--328, 1982

    1982

  71. [79]

    Yamaguchi

    A. Yamaguchi. Morava K -theory of double loop spaces of spheres. Mathematische Zeitschrift , 199(4):511--523, 1988

    1988

  72. [80]

    The power operation structure on M orava E -theory of height 2 at the prime 3

    Yifei Zhu. The power operation structure on M orava E -theory of height 2 at the prime 3. Algebr. Geom. Topol. , 14(2):953--977, 2014

    2014

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