/pdf/on-the-cohomology-of-spaces-with-two-non-vanishing-homotopy-6b1lwv20fi.pdf

On the stable cohomology groups of certain Postnikov complexes

On the Cohomology of Spaces with Two Non-Vanishing Homotopy Groups.

https://projecteuclid.org/download/pdf_1/euclid.jmsj/1261147293

Homologie Singuliere Des Espaces Fibres

On sait que les complexes K (/7, q) introduits par Eilenberg-MacLane dans [4] jouent un r61e essentiel dans un grand nombre de questions de topologie alg6brique. Le pr6sent article est une contribution ~ leur 6tude. En nous appuyant sur un th~or~me d6montr6 par A. Borel dans sa th~se [2], nous d~terminons les alg~bres de cohomologie modulo 2 de ces complexes, tout au moins lorsque le g roupe / /poss~de un nombre fini de g~n~rateurs. Ceci fair l 'objet du w 2. Dans le w 3 nous 6tudions le comportement asymptotique des s6ries de Poincar6 des alg~bres de cohomologie pr6cddentes ; nous en ddduisons que, lorsqu'un espace X v~rifie des conditions tr~s larges (par exemple, lorsque X est un poly~dre fini, simplement connexe, d'homologie modulo 2 non triviale), il existe une infinit6 d'entiers i tels que le groupe d'homotopie gi(X) contienne un sous-groupe isomorphe ~ Z ou ~ Z2. Dans le w 4 nous pr~cisons les relations qui lient les complexes K(II, q) et les diverses ((operations cohomologiques)>; ceci nous fournit notamment une m6thode permettant d'dtudier les relations entre i-carr6s it~rds. Le w 5 contient le calcul des groupes ~.+3(S~) et g.+4(S.); ce calcul est effectud en combinant les r6sultats des w167 2 et 4 avec ceux d'une Note de H. Cartan et l 'auteur ([3], voir aussi [14]). Les w167 4 et 5 sont inddpendants du w 3. Les principaux rdsultats de cet article ont gt6 rdsumds dans une Note aux Comptes Rendus [9].

Cohomologie modulo 2 des complexes d’Eilenberg-MacLane

1 Introduction 2 The homotopy groups 3 The classical theorems of homotopy theory 4 The exact homotopy sequence 5 Fibre-Spaces 6 The Hopf invariant and suspension theorems 7 Whitehead cell-complexes 8 Homotopy groups of complexes

An Introduction to Homotopy Theory

https://www.pnas.org/content/pnas/38/8/720.full.pdf

The Iteration of the Steenrod Squares in Algebraic Topology.

For the context of this question, see [5] (especially pp. 436-438), [4, Chapter VI] and [6, §§20, 21]. This theorem results from reasonings with secondary cohomology operations. I t is generally understood that a secondary operation corresponds to a relation between primary operations. One may formalize the notion of a \"relation\" by introducing pairs (d, z), algebraic in nature, as follows. Let p be a prime; let A be the Steenrod algebra [2, p. 43] over Zp. One defines the notion of a graded left module M over the graded algebra A so that M= ^2q Mq and AqMr(ZMq+r. For example, let us write H«(X) for H*(X; Zp), H*(X) for £ , H*(X; Z9) and H+(X) for ]£«>o H*(X; Zp); then H*(X) and H+(X) are graded left modules over A. Let M, N be such modules; one defines the notion of an A -map ƒ : M->N of degree r so that f(Mq) CNq+r. A pair (d, 2), then, is to have the following nature. The first entry d is to be an A -map d: G—>C0 of degree zero. Here Co, G are to be modules in the above sense; we require, moreover, that they are locally finitely-generated and free, and that (Ci)q — 0 if q<i (i = 0, 1). The second entry z is to be a homogeneous element of Ker d. Let (d, z), then, be a pair of this sort. We call $ a stable secondary cohomology operation associated with (J, 2), if it satisfies the following axioms. AXIOM (1). $(e) is defined for each A-map e: C0-^H (X) of degree w ^ l and such that ed = 0. Such a map € is determined by its values on the elements of an ^4-base of C0. I t therefore corresponds to a set of elements of H (X). In particular, if Co is free on one given generator c, we write u~ec; we may thus consider $ as a function of one variable u, where u runs over a subset of H(X). In this case we Write $(u) for $(€). For the next axiom, set deg(z)=n + l, l e t / : Ci-*H(X) run over

/pdf/on-the-nonexistence-of-elements-of-hopf-invariant-one-49lmr0tf1g.pdf

On the stable cohomology groups of certain Postnikov complexes

Citations

On the Cohomology of Spaces with Two Non-Vanishing Homotopy Groups.

References

Homologie Singuliere Des Espaces Fibres

Cohomologie modulo 2 des complexes d’Eilenberg-MacLane

An Introduction to Homotopy Theory

The Iteration of the Steenrod Squares in Algebraic Topology.

On the nonexistence of elements of Hopf invariant one

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