{
  "id": "math/0201134",
  "version": "v1",
  "published": "2002-01-15T15:51:20.000Z",
  "updated": "2002-01-15T15:51:20.000Z",
  "title": "On the cohomology algebra of a fiber",
  "authors": [
    "Luc Menichi"
  ],
  "comment": "Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-36.abs.html",
  "journal": "Algebr. Geom. Topol. 1 (2001) 719-742",
  "categories": [
    "math.AT"
  ],
  "abstract": "Let f:E-->B be a fibration of fiber F. Eilenberg and Moore have proved that there is a natural isomorphism of vector spaces between H^*(F;F_p) and Tor^{C^*(B)}(C^*(E),F_p). Generalizing the rational case proved by Sullivan, Anick [Hopf algebras up to homotopy, J. Amer. Math. Soc. 2 (1989) 417--453] proved that if X is a finite r-connected CW-complex of dimension < rp+1 then the algebra of singular cochains C^*(X;F_p) can be replaced by a commutative differential graded algebra A(X) with the same cohomology. Therefore if we suppose that f:E-->B is an inclusion of finite r-connected CW-complexes of dimension < rp+1, we obtain an isomorphism of vector spaces between the algebra H^*(F;F_p) and Tor^{A(B)}(A(E),F_p) which has also a natural structure of algebra. Extending the rational case proved by Grivel-Thomas-Halperin [PP Grivel, Formes differentielles et suites spectrales, Ann. Inst. Fourier 29 (1979) 17--37] and [S Halperin, Lectures on minimal models, Soc. Math. France 9-10 (1983)] we prove that this isomorphism is in fact an isomorphism of algebras. In particular, $H^*(F;F_p) is a divided powers algebra and p-th powers vanish in the reduced cohomology \\mathaccent \"707E {H}^*(F;F_p).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-01-15T15:51:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55R20",
      "55P62",
      "18G15",
      "57T30",
      "57T05"
    ],
    "keywords": [
      "cohomology algebra",
      "vector spaces",
      "rational case",
      "natural isomorphism",
      "hopf algebras"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}