## A general algebraic approach to Steenrod operations

Citations: | 46 - 3 self |

### BibTeX

@MISC{May_ageneral,

author = {J. Peter May},

title = {A general algebraic approach to Steenrod operations},

year = {}

}

### Years of Citing Articles

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### Citations

68 |
Cohomology Operations
- Epstein
- 1962
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Citation Context ... H~(X) is a left module over the opposite algebra of the Steenrod algebra). The following proposition was used in the proof of Lemma 4.6. Formula (2) of the proof is essentially Steenrod's definition =-=[30]-=- of the D.. 1 Proposition 9. t. Let X be a space andlet d = ~(t@D):W@wCt(X )-> W%Cg,(X) p. Consider d.:H,(~;H~(X)),. . o --> H-'-(w;H,',(x)P)'.,. , Let x ~ Hs(X). Then (i) If p = 2, d l~(er~X ) = ~er+... |

30 |
Homology of iterated loop spaces
- Dyer, Lashoff
- 1962
(Show Context)
Citation Context ... particular, tp = tZm+l = (-1)mq'm~ a p (since each i.j = 0). Now let j = (Zs-q+l)(p-1) and define chains c and c' in W~ N p bythe following rr formulas (where, by convention, e. = 0 if i < 0): 1 m m =-=(6)-=- )kej_Zk~ c = k=OE (-I - E (-1)k ej+l_Zk~(a-1 tZk+l k=l 1)P'Z - tZk; (7) ITI k c' = E(-1) ej_l.ZkOtZk+l + E (-1)k ej_zk~tZk k=0 k=l m Then an easy computation, which uses Definitim 1.2 and (3), gives ... |

25 | On the structure and applications of the Steenrod algebra - Adams - 1958 |

18 | Cohomologie modulo 2 des compléxes d’Eilenberg–MacLane - Serre - 1953 |

15 |
Über die Steenrodschen Kohomologieoperationen
- Dold
- 1961
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Citation Context ...uction, which uses the explicit formula for S , yields- I 6 7 - (4) tZk = E (-1)kq(k-1)' bilaZbiZaZ''" bikaZ , 1 _< k_< m, summed over all I k-tuples I = (i I ..... ik) such that E ij = p-Zk ; and • =-=(5)-=- tZk+l = E(-1)kqklb it 2 , hlkaZblk+l a .. a, 0<_k< m, summedover ali I o (k+l)-tuples I = (i 1 . . . . . ik+l) such that ~ ij = p-l-Zk. In particular, tp = tZm+l = (-1)mq'm~ a p (since each i.j = 0).... |

8 |
The relations on Steenrod powers of cohomology classes. Algebraic geometry and topology, A symposium in honor of S
- Adem
- 1957
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Citation Context ...ite W(p,A) for W. Of course,- 158 - W(p,A) = W(p, Z) ~A. The structure of W(p, Zp) shows that H.(~r; Zp) = H(W(p, Zp)~rrZp) is given, with its Bockstein operation ~3 and coproduct ~b, by the formula =-=(2)-=- Ho.(~r;Zp).,. has Zp-basis {ell i>0}_ suchthat ~3(e2i)= ezi_l and ¢(e i) = E e.j ~ e k if p = 2 or i is odd, J2(e2i ) = E e2j(~e2k j+k = i j+k = i We embed ~ in E by o~(1 ..... p) = (p, 1 ..... p-l),... |

7 | Remarks on a theorem of
- unknown authors
- 1964
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Citation Context ...ion 2. t, and the resulting composite is naturally A~r-homotopic to the original @ defined in terms of W, @ satisfies condition (ii) of Definition Z. 1. By the lemma, formula (11) specializes to give =-=(12)-=- @(eo@X ) = D*~(x) for any x e B*(C) r and (t3) @(w(~ x) * B°' *(C) r = ~(w)D ~(x) for any xe and w ~ W. By (t0) and (12), @ satisfies condition (i) of Definition 2. t. Since @ is natural on morphisms... |

6 |
Homology operations and loop spaces
- Browder
- 1960
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Citation Context ...= I, an easy calculation demonstrates that (3) d(tl) = to , d(tzk) = (~-I 1)tZk_1 and d(t2k+1) = Ntzk , l<__k<_m. A straightforward induction, which uses the explicit formula for S , yields- I 6 7 - =-=(4)-=- tZk = E (-1)kq(k-1)' bilaZbiZaZ''" bikaZ , 1 _< k_< m, summed over all I k-tuples I = (i I ..... ik) such that E ij = p-Zk ; and • (5) tZk+l = E(-1)kqklb it 2 , hlkaZblk+l a .. a, 0<_k< m, summedover... |

6 | On the structure of Hopf Algebras Ann - Milnor, Moore - 1965 |

3 |
Primary cohomology operations for simplicial Lie algebras
- Priddy
- 1970
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Citation Context ...(L/M)) as an object of ~,,, with diagonal D = ~b , the reduced coproduct as defined in Remarks 7.4(ii). Thus Theorem 7.9 applies directly, and pO = 0 follows from the previous lemma and corollary. In =-=[22]-=-, Priddy has given a different definition of H, (g) and H~'~(L) Let W be the functor from simplicial Z -algebras to ~(G defined by Moore [20]. P is a simplicial Z -algebra, then %(A) = Zp and Wq(A) = ... |

2 | Cohomology operations in iterated loop spaces - Nishida |

1 |
Topology of H -spaces and H-squaring operations. n
- Araki, Kudo
- 1956
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Citation Context ...I. for 0 <_ i <_ p by the inductive formula Define t.e (a,b) p i (Z) t be; t 1 b p-1 = S(¢~ -1 o = = a', tZk t2k_l - t2k_l ) ; tZk+l = S(Nt2k ) Since dS +Sd = I, an easy calculation demonstrates that =-=(3)-=- d(tl) = to , d(tzk) = (~-I 1)tZk_1 and d(t2k+1) = Ntzk , l<__k<_m. A straightforward induction, which uses the explicit formula for S , yields- I 6 7 - (4) tZk = E (-1)kq(k-1)' bilaZbiZaZ''" bikaZ ,... |

1 |
Quelques proprietes des produits de
- Paris
- 1955
(Show Context)
Citation Context ... )kej_Zk~ c = k=OE (-I - E (-1)k ej+l_Zk~(a-1 tZk+l k=l 1)P'Z - tZk; (7) ITI k c' = E(-1) ej_l.ZkOtZk+l + E (-1)k ej_zk~tZk k=0 k=l m Then an easy computation, which uses Definitim 1.2 and (3), gives =-=(8)-=- d(c) = e.@b p J and d(c') = -ej_l~)bP (j = (ZS - q+l)(p-1)) In calculating d(c), the salient observations are that Nt = 0, that P ae.~t = e.~a-lt for t e K p by the very definition of a tensor produc... |

1 |
Lie Algebras. Interscience Publishers
- Jacobson, Kochman
- 1962
(Show Context)
Citation Context ... In calculating d(c), the salient observations are that Nt = 0, that P ae.~t = e.~a-lt for t e K p by the very definition of a tensor product, and that 1 1 (a -1- 1) p-1 = N in Z w. Finally, define P =-=(9)-=- Ps(a) = (-1)Sv(q-1)O(e) and ~Ps(a) = (-1)Su(q-1)0(e ') . If a is a cycle, b = 0, then t. = 0 for i< p and t = (-1)mqm 'a p, hence 1 p (i0) c = (-l)m(q+l)m'e(zs_q)(p_l)@a p and c' = (-1)m(cl+l)m.'e(zs... |

1 |
cohomology operations. The factorization of cyclic reduced powers by secondary Mere
- Liulevicius
- 1962
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Citation Context ...restricted Lie algebras. The theory is applied to the cohomology of cocommutative Hopf algebras in section II; the operations here are important in the study of the cohomology of the Steenrod algebra =-=[13, 18]-=-. The present analysis arose out of work on iterated loop spaces, but this application will appear elsewhere. The material of sections 6 and 9, which is peripheral to the study of Steenrod operations,... |

1 |
Constructions sur les complexes d'anneaux. Seminaire ri Cartan
- Moore
- 1954
(Show Context)
Citation Context ...d pO = 0 follows from the previous lemma and corollary. In [22], Priddy has given a different definition of H, (g) and H~'~(L) Let W be the functor from simplicial Z -algebras to ~(G defined by Moore =-=[20]-=-. P is a simplicial Z -algebra, then %(A) = Zp and Wq(A) = Aq i (~" ~A p - "" o' q > 0, as Z -modules. P The face and degeneracy operators are as defined in [15, p. 87]. For L ~ t~ , WV(L) is a simpli... |

1 |
groups of symmetric groups and reduced power
- Homology
- 1953
(Show Context)
Citation Context ...the cohomology of simplicial restricted Lie algebras, in the cohomology of cocommutative Hopf algebras, and in the homology of infinite loop spaces (where they were introduced mod 2 by Araki and Kudo =-=[3]-=- and mod p, p>2, by Dyer and Lashof [6]). The purpose of this expository paper is to develop a general algebraic setting in which all such operations can be studied simultaneously. This approach allow... |