## Problems in the Steenrod algebra (1998)

Venue: | Bull. London Math. Soc |

Citations: | 21 - 1 self |

### BibTeX

@ARTICLE{Wood98problemsin,

author = {R. M. W. Wood},

title = {Problems in the Steenrod algebra},

journal = {Bull. London Math. Soc},

year = {1998},

volume = {30},

pages = {449--517}

}

### OpenURL

### Abstract

This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which non-specialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below. Contents 1 Historical background 4

### Citations

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Citation Context ... 2-primary part of the stable homotopy groups of spheres. In another direction, Thom [200, 197] linked bordism theory of manifolds and characteristic classes of vector bundles to the Steenrod algebra =-=[134]-=-. The internal structure of A was investigated by Adams in section 5 of [2]. He proved, in particular, that any finite collection of elements in A generates a finite subalgebra. In fact, A is a local ... |

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Symmetric functions and Hall polynomials. Second edition. With contributions by A. Zelevinsky
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Citation Context ...ative chapters 3son the Steenrod algebra, and likewise the books of Adem and Milgram [12] and Benson [23] on the cohomology of groups. A standard reference for symmetric functions is Macdonald's book =-=[121]-=-, and the representation theory of symmetric groups and general linear groups over finite fields is treated in the book of James and Kerber [90]. The differential operator approach to the Steenrod alg... |

417 |
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Citation Context ... reference for symmetric functions is Macdonald's book [121], and the representation theory of symmetric groups and general linear groups over finite fields is treated in the book of James and Kerber =-=[90]-=-. The differential operator approach to the Steenrod algebra and the Landweber-Novikov algebra touches on certain combinatorial material which can be found in standard texts such as Aigner [13], Camer... |

346 |
Combinatorial enumeration
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Citation Context ...e Steenrod algebra and the Landweber-Novikov algebra touches on certain combinatorial material which can be found in standard texts such as Aigner [13], Cameron [37], Comtet [47], Goulden and Jackson =-=[77]-=- and Henrici [80], as well as the classic text of MacMahon [122]. Recent work on the Steenrod algebra and the Landweber-Novikov algebra in relation to combinatorics can be found in [163, 164, 165, 169... |

295 | Hopf algebras and their actions on rings - Montgomery - 1993 |

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Citation Context ...lgebra impinges on areas of mathematics concerning D-modules, Lie algebras, deformation of Hopf algebras, quantum groups and the Weyl algebra. Relevant information can be found in a number of sources =-=[1, 27, 48, 76, 97, 124, 125, 127, 129, 144, 194]-=-. The present article expands on material found in [221] but stems originally from a talk entitled "Facts and Fancies in the Steenrod Algebra" delivered during the topology conference at G"ottingen in... |

204 |
Complex cobordism and stable homotopy groups of spheres
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Citation Context ...n the Steenrod algebra. The main thrust of research on the Steenrod algebra has naturally been concerned with modules over A, since this lies at the heart of algebraic topology. The bibliographies of =-=[125, 145, 162, 189]-=- and [170] give some indication of past work and current progress on the Steenrod algebra and its applications to homotopy theory. The present article, on the other hand, has the very limited goal of ... |

144 |
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Citation Context ...motopy theory. The integral approach to Steenrod squares is closely related to operations in complex bordism theory, as Landweber noted in his original paper [111]. The Chicago lecture notes of Adams =-=[4]-=- describe Novikov's work on this subject, and a general survey of bordism theory can be found in Stong [190]. More recent references are Ravenel [162] and Kochman [101]. Ever since its inception, the ... |

107 |
Foundations of Quantum Group Theory,” Cambridge Univ
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Citation Context ...lgebra impinges on areas of mathematics concerning D-modules, Lie algebras, deformation of Hopf algebras, quantum groups and the Weyl algebra. Relevant information can be found in a number of sources =-=[1, 27, 48, 76, 97, 124, 125, 127, 129, 144, 194]-=-. The present article expands on material found in [221] but stems originally from a talk entitled "Facts and Fancies in the Steenrod Algebra" delivered during the topology conference at G"ottingen in... |

103 |
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Citation Context ...(X) which were later [225, 226] seen to be related, via Kronecker duality, to O/(Sqk), where O/ is a certain conjugation operation in the Steenrod algebra introduced by Thom in his study of manifolds =-=[200]-=-. Serre [171] showed that the Steenrod squares generate all stable operations in the cohomology theory. From a topological point of view, A is the algebra of stable operations of H\Lambdasover F2 gene... |

95 |
Quantum groups, Graduate texts in mathematics 155
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Citation Context ...lgebra impinges on areas of mathematics concerning D-modules, Lie algebras, deformation of Hopf algebras, quantum groups and the Weyl algebra. Relevant information can be found in a number of sources =-=[1, 27, 48, 76, 97, 124, 125, 127, 129, 144, 194]-=-. The present article expands on material found in [221] but stems originally from a talk entitled "Facts and Fancies in the Steenrod Algebra" delivered during the topology conference at G"ottingen in... |

71 |
Polynomial invariants of finite groups
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- 1995
(Show Context)
Citation Context ... are Ravenel [162] and Kochman [101]. Ever since its inception, the Steenrod algebra has been allied to group cohomology theory and invariant theory [188]. The book on invariant theory by Larry Smith =-=[183]-=- contains informative chapters 3son the Steenrod algebra, and likewise the books of Adem and Milgram [12] and Benson [23] on the cohomology of groups. A standard reference for symmetric functions is M... |

69 | On the non-existence of elements of Hopf invariant one
- Adams
- 1960
(Show Context)
Citation Context ...bra by Sq0 and the Sq2 k for k * 0. In the decade 1950-1960, the Steenrod algebra became one of the most powerful tools in algebraic topology. For example, Adams solved the Hopf invariant one problem =-=[3]-=-, thereby proving that non-singular bilinear maps Rn \ThetasRn ! Rn exist only for dimensions n = 1; 2; 4; 8, where they are realised by real, complex, quaternionic and Cayley multiplication. The fact... |

69 |
Cohomology Operations
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- 1962
(Show Context)
Citation Context ...n the Steenrod algebra. The main thrust of research on the Steenrod algebra has naturally been concerned with modules over A, since this lies at the heart of algebraic topology. The bibliographies of =-=[125, 145, 162, 189]-=- and [170] give some indication of past work and current progress on the Steenrod algebra and its applications to homotopy theory. The present article, on the other hand, has the very limited goal of ... |

69 |
Notes on Cobordism Theory
- Stong
(Show Context)
Citation Context ... between the three products .; ffl; ffi. This problem has been treated in a topological context in terms of Chern classes, Stiefel-Whitney classes, cohomology of classifying spaces and Thom complexes =-=[19, 190]-=-. However, it would be interesting to codify the material in terms of differential operators. The three conjugations O/ = O/.; O/ffl; O/ffi all have the same value O/(Dk) = \Gamma Dk on the primitives... |

68 |
Elementary proofs of some results of cobordism theory using Steenrod operations
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- 1971
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Citation Context ...th theory [29, 200, 225], double point sets in bordism with singularities, and SpanierWhitehead duality [130]. Dold sets up the Steenrod algebra for cohomology of a topological space in [60]. Quillen =-=[161]-=- uses Steenrod operations in bordism, further developed by Tom Dieck [202], in connection with formal groups and the Lazard ring. In [25], Bisson and Joyal treat similar topics in terms of a divided d... |

63 |
An introduction to noncommutative noetherian rings
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- 1989
(Show Context)
Citation Context |

44 |
Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture
- Schwartz
- 1994
(Show Context)
Citation Context ...he main thrust of research on the Steenrod algebra has naturally been concerned with modules over A, since this lies at the heart of algebraic topology. The bibliographies of [125, 145, 162, 189] and =-=[170]-=- give some indication of past work and current progress on the Steenrod algebra and its applications to homotopy theory. The present article, on the other hand, has the very limited goal of addressing... |

43 |
Representations and cohomology II: Cohomology of groups and modules, volume 31 of Cambridge studies in advanced mathematics
- Benson
- 1991
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Citation Context ...heory and invariant theory [188]. The book on invariant theory by Larry Smith [183] contains informative chapters 3son the Steenrod algebra, and likewise the books of Adem and Milgram [12] and Benson =-=[23]-=- on the cohomology of groups. A standard reference for symmetric functions is Macdonald's book [121], and the representation theory of symmetric groups and general linear groups over finite fields is ... |

42 |
A history of Algebraic and Differential Topology
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- 1989
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Citation Context ...od algebra [187] at various primes, and the contributions of the early pioneers to the development of the Steenrod algebra are expounded in Dieudonn'e's history of algebraic and differential topology =-=[59]-=-. The Steenrod algebra is a graded Hopf algebra, for which the papers of Milnor [131] and Milnor and Moore [133] are standard references. The differential operator approach to the Steenrod algebra imp... |

42 |
Incidence Hopf algebras
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- 1994
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Citation Context ...en and Jackson [77] and Henrici [80], as well as the classic text of MacMahon [122]. Recent work on the Steenrod algebra and the Landweber-Novikov algebra in relation to combinatorics can be found in =-=[163, 164, 165, 169]-=-. Steenrod's original methods of defining the Steenrod algebra [187] at various primes, and the contributions of the early pioneers to the development of the Steenrod algebra are expounded in Dieudonn... |

40 |
Algebraic D-modules
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Citation Context |

37 |
A fundamental system of invariants of the general modular linear group with a solution of the form problem
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- 1911
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Citation Context ...currences of the trivial representation in the polynomial algebra can be identified with the ring of invariants of the general linear group GL(n; F2). This is known classically as the Dickson algebra =-=[58, 123, 205, 214]-=-, and is a polynomial subring of W(n) \OmegasF2 generated by the Dickson invariants. The action of the Steenrod squares on the Dickson invariants and, more generally, invariants of parabolic subgroups... |

28 |
Cohomology operations and applications in homotopy theory
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- 1968
(Show Context)
Citation Context ...n the Steenrod algebra. The main thrust of research on the Steenrod algebra has naturally been concerned with modules over A, since this lies at the heart of algebraic topology. The bibliographies of =-=[125, 145, 162, 189]-=- and [170] give some indication of past work and current progress on the Steenrod algebra and its applications to homotopy theory. The present article, on the other hand, has the very limited goal of ... |

26 |
Hopf algebras of symmetric functions and class functions, Combinatoire et Combinatoire et représentation du groupe symétrique (Actes Table
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- 1976
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Citation Context ...the algebraic Thom map. Example 3.1 OE(D(K)) = m(K); OE(Dk) = pk; OE(SQr) = er; OE ` D ffir1 r! ' = hr: We recall that the Dk are the primitives in the Hopf algebra D. In \Lambdasthere is a coproduct =-=[71, 121, 208]-=- which makes \Lambdasinto a Hopf algebra with respect to the dot product. The primitives are the pk. We noted that OE(Dk) = pk, and it can be checked that the algebraic Thom map is a coalgebra map. He... |

24 |
Cohomology of finite groups, Grundlehren der Mathematischen Wissenschaften, 309
- Adem, Milgram
- 1994
(Show Context)
Citation Context ...oup cohomology theory and invariant theory [188]. The book on invariant theory by Larry Smith [183] contains informative chapters 3son the Steenrod algebra, and likewise the books of Adem and Milgram =-=[12]-=- and Benson [23] on the cohomology of groups. A standard reference for symmetric functions is Macdonald's book [121], and the representation theory of symmetric groups and general linear groups over f... |

24 |
Methods of algebraic topology from the point of view of cobordism theory
- Novikov
- 1967
(Show Context)
Citation Context ... related to the Steenrod algebra Milnor [132] introduced the complex analogue MU\Lambdasof Thom's bordism theory. The stable operations MU\Lambda (M U ) were worked out by Landweber [111] and Novikov =-=[149]-=-. Briefly, MU\Lambda (M U ) can be identified additively with MU\Lambda (pt) \OmegasS\Lambda , where MU\Lambda (pt) is the value of MU\Lambdason a point, and S\Lambdasis the LandweberNovikov algebra. ... |

24 |
Products of cocycles and extensions of mappings
- Steenrod
- 1947
(Show Context)
Citation Context ...lationship between the differential operator algebra and the Landweber-Novikov algebra. Part of the present work has been supported by EPSRC grant GR/K 05856. 1 Historical background In 1947 Steenrod =-=[186]-=- introduced certain linear operators of ordinary cohomology theory H\Lambdasover F2 defined in terms of cocycles in a simplicial cochain complex by modifying the Alexander- ^Cech-Whitney formula for t... |

23 |
The iteration of the Steenrod squares in algebraic topology
- Adem
(Show Context)
Citation Context ...mposition, subject to relations which vanish on the cohomology of all spaces X. Wu [224, 222] conjectured certain relations among the squaring operations, which were proved by Adem. Theorem 1.2 (Adem =-=[10, 11]-=-) All relations in the Steenrod algebra are generated by the set of Adem relations SqiSqj = X 0^k^[i=2]sj \Gammask \Gammas1 i \Gammas2k !Sq i+j\Gamma kSqk for 0 ! i ! 2j, where [i=2] denotes the great... |

19 |
Finite H-spaces and algebras over the Steenrod algebra
- Adams, Wilkerson
- 1980
(Show Context)
Citation Context ...e Dickson invariants. The action of the Steenrod squares on the Dickson invariants and, more generally, invariants of parabolic subgroups of general linear groups, is worked out in a number of places =-=[9, 81, 83, 99, 146, 193, 214]-=-, and global product formulae are produced in [116]. The restricted hit problem for the Dickson algebra is solved for a small number of variables in [72, 84, 87, 88], but the general problem appears d... |

18 |
Cohomologie modulo 2 des complexes d’Eilenberg-MacLane
- Serre
- 1953
(Show Context)
Citation Context ...e later [225, 226] seen to be related, via Kronecker duality, to O/(Sqk), where O/ is a certain conjugation operation in the Steenrod algebra introduced by Thom in his study of manifolds [200]. Serre =-=[171]-=- showed that the Steenrod squares generate all stable operations in the cohomology theory. From a topological point of view, A is the algebra of stable operations of H\Lambdasover F2 generated by the ... |

17 |
Combinatorial theory, Grundlehren der Mathematischen Wissenschaften
- Aigner
- 1979
(Show Context)
Citation Context ...Kerber [90]. The differential operator approach to the Steenrod algebra and the Landweber-Novikov algebra touches on certain combinatorial material which can be found in standard texts such as Aigner =-=[13]-=-, Cameron [37], Comtet [47], Goulden and Jackson [77] and Henrici [80], as well as the classic text of MacMahon [122]. Recent work on the Steenrod algebra and the Landweber-Novikov algebra in relation... |

17 |
Generic representations of the finite general linear groups and the Steenrod algebra
- Kuhn
(Show Context)
Citation Context ...ed more extensively in Li's work [116]. The global approach is now usually adopted in setting up the Steenrod algebra in an algebraic context. For example, in Larry Smith's book [183] and Kuhn's work =-=[107, 108, 109]-=-, the rules in Lemma 1.3 are extended to cover finite fields through generating functions for total operations. Let Fq denote the Galois field, where q is a power of the prime p. The Steenrod reduced ... |

15 |
Über die Steenrodschen Kohomologieoperationen
- Dold
- 1961
(Show Context)
Citation Context ...relates to Smith theory [29, 200, 225], double point sets in bordism with singularities, and SpanierWhitehead duality [130]. Dold sets up the Steenrod algebra for cohomology of a topological space in =-=[60]-=-. Quillen [161] uses Steenrod operations in bordism, further developed by Tom Dieck [202], in connection with formal groups and the Lazard ring. In [25], Bisson and Joyal treat similar topics in terms... |

14 |
Cobordism operations and Hopf algebras
- Landweber
- 1967
(Show Context)
Citation Context ... in the general context of homology and homotopy theory. The integral approach to Steenrod squares is closely related to operations in complex bordism theory, as Landweber noted in his original paper =-=[111]-=-. The Chicago lecture notes of Adams [4] describe Novikov's work on this subject, and a general survey of bordism theory can be found in Stong [190]. More recent references are Ravenel [162] and Kochm... |

12 |
The Landweber–Novikov algebra and formal vector fields on the line, Funct
- Buchstaber, Shokurov
- 1978
(Show Context)
Citation Context ...iables xi [5, 215]. For interpretations of Landweber-Novikov operations as differential operators in the context of conformal field theory, quantum groups and diffeomorphisms of the line, we refer to =-=[33, 34, 35, 98, 150]-=-. It would also be interesting to know if the complex bordism of a compact Hausdorff topological space X could be defined in an algebraic manner, perhaps in terms of the algebra of complex-valued func... |

12 |
Reduced powers of cohomology classes
- Steenrod
- 1952
(Show Context)
Citation Context .... Recent work on the Steenrod algebra and the Landweber-Novikov algebra in relation to combinatorics can be found in [163, 164, 165, 169]. Steenrod's original methods of defining the Steenrod algebra =-=[187]-=- at various primes, and the contributions of the early pioneers to the development of the Steenrod algebra are expounded in Dieudonn'e's history of algebraic and differential topology [59]. The Steenr... |

12 |
Espaces fibrés en sphères et carrés de
- Thom
- 1952
(Show Context)
Citation Context ... is so called because a complex n-plane bundle E over a connected compact Hausdorff base space X is induced from a certain universal bundle over BU (n) by a map of X into BU (n). The Thom isomorphism =-=[134, 199]-=- OE: H\Lambda (X) ! ~H\Lambda +n(E\Lambda ) is an isomorphism from the cohomology of X to the reduced cohomology of the one-point compactification E\Lambdasof E, known as the Thom complex of E. For th... |

11 |
On the Adem relations, Topology 21
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- 1982
(Show Context)
Citation Context ...s in these areas can be traced through the work of Crabb, Crossley and Hubbuck [49, 50, 51, 52, 53], Ken Monks [139, 138, 142, 143, 140], and Bill Singer and Judith Silverman [174, 175, 178, 182]. In =-=[36]-=-, Bullett-Macdonald devise a method for generating the Adem relations by equating coefficients in certain products of formal power series. This global approach to the study of squaring operations via ... |

10 |
l'iteration des operations de
- Sur
- 1955
(Show Context)
Citation Context ... grading and never increase length of monomials when the re-write rules are applied to SqiSqj for 0 ! i ! 2j. In his work on the cohomology of Eilenberg-MacLane spaces for the group of order 2, Serre =-=[171, 44]-=- gave a method of deriving the Adem relations in terms of a faithful representation of A on the cohomology of the infinite product of infinite real projective spaces. Let W = Z[x1; : : : ; xn; : : :] ... |

10 |
stable homotopy and Adams spectral sequences
- Kochman, Bordism
- 1996
(Show Context)
Citation Context ...he Chicago lecture notes of Adams [4] describe Novikov's work on this subject, and a general survey of bordism theory can be found in Stong [190]. More recent references are Ravenel [162] and Kochman =-=[101]-=-. Ever since its inception, the Steenrod algebra has been allied to group cohomology theory and invariant theory [188]. The book on invariant theory by Larry Smith [183] contains informative chapters ... |

9 | Codimension one immersions and the Kervaire invariant one problem
- Eccles
(Show Context)
Citation Context ...olynomials divisible by x1 \Deltas\Deltas\Deltasxn. Closed formulae for these actions have been of interest to algebraic and differential topologists, for example in the immersion theory of manifolds =-=[64, 65, 66, 68]-=-. Classically, the Wu formulae [134, 223] answer the question of how the Steenrod squares act on the elementary symmetric polynomials in \Lambda (n) \OmegasF2: Sqk(wm) = wkwm + `k \Gammasm1 'wk\Gamma ... |

9 |
Lie algebras. Interscience Tracts
- Jacobson
- 1962
(Show Context)
Citation Context ...ll gradings. A rational basis for N , which is the same as D and U over Q by Theorem 2.5, is given by the classical Poincar'eBirkhoff-Witt theorem in the universal enveloping algebra of a Lie algebra =-=[89]-=-. The set of composites Dffir1k1 ffi Dffir2k2 ffi \Deltas\Deltas\Deltasffi Dffiraka , where k1 ? k2 ? : : : ? ka, forms an additive basis of U . It is not immediately clear that a similar statement is... |

8 |
The relations on Steenrod powers of cohomology classes. Algebraic geometry and topology, A symposium in honor of S
- Adem
- 1957
(Show Context)
Citation Context ...mposition, subject to relations which vanish on the cohomology of all spaces X. Wu [224, 222] conjectured certain relations among the squaring operations, which were proved by Adem. Theorem 1.2 (Adem =-=[10, 11]-=-) All relations in the Steenrod algebra are generated by the set of Adem relations SqiSqj = X 0^k^[i=2]sj \Gammask \Gammas1 i \Gammas2k !Sq i+j\Gamma kSqk for 0 ! i ! 2j, where [i=2] denotes the great... |

8 |
Modular representations on the homology of powers of real projective space, from: “Algebraic topology
- Boardman
- 1991
(Show Context)
Citation Context ...lem asks for a description of C(n) 58sas an M (n; F2)-module. Information in the three-variable case can be found in [42, 50, 91], and progress on the general problem can be traced through the papers =-=[15, 26, 50, 178, 179, 182, 218, 219]-=-. Problems still exist in four or more variables. The following results illustrate some of the progress. Recall the numerical function _(d) as in Definition 4.4 as the least number k for which it is p... |

8 |
Truncated symmetric powers and modular representations of GLn
- Doty, Walker
- 1996
(Show Context)
Citation Context ...se known as the classifying space BT n of the n-torus [39, 67, 79, 135, 136, 216, 217]. Little is known about the odd prime analogue of the first occurrence problem, although a few cases are resolved =-=[40, 41, 62, 216]-=-. The first occurrence of a simple module as a submodule is also an interesting question, especially in conjunction with the problem of linking it to the first occurrence as composition factor via Ste... |

8 |
Characteristic numbers of immersions and self-intersection manifolds
- Eccles
- 1993
(Show Context)
Citation Context ...olynomials divisible by x1 \Deltas\Deltas\Deltasxn. Closed formulae for these actions have been of interest to algebraic and differential topologists, for example in the immersion theory of manifolds =-=[64, 65, 66, 68]-=-. Classically, the Wu formulae [134, 223] answer the question of how the Steenrod squares act on the elementary symmetric polynomials in \Lambda (n) \OmegasF2: Sqk(wm) = wkwm + `k \Gammasm1 'wk\Gamma ... |

8 |
Sur la structure des A-modules instables injectifs, Topology 28
- Lannes, Schwartz
- 1989
(Show Context)
Citation Context ...eyl modules for the general linear groups. Information can be found in [69] for the relationship between the first occurrence problems and Lannes' theory of unstable modules over the Steenrod algebra =-=[107, 108, 109, 112, 113]-=-. We can ask for an analogue, for the differential operator algebra, of the AdamsGunawardena-Miller result [7] which states that all grade-preserving linear transformations of the polynomial algebra W... |

8 |
The transfer in homological algebra
- Singer
- 1989
(Show Context)
Citation Context ...eterson [158]. Several reasons for studying the hit problem are listed in [219]. These include Frank Peterson's work on Stiefel-Whitney classes and bordism of manifolds [158, 156], Bill Singer's work =-=[181]-=- on covariants of the general linear group GL(n; F2) with applications to the Adams spectral sequence, and the relationship between the modular representation theory of the general linear groups over ... |