On the cohomology algebra of a fiber. (English) Zbl 0981.55006

For a fibration \(f:E\rightarrow B\) of fiber \(F\) and simple connected base \(B\) , the well-known Eilenberg-Moore isomorphism \(H^{*}(F;{\mathbf k})\cong Tor^{C^{*}(B)}(C^{*}(E);{\mathbf k})\) is an isomorphism of graded vector spaces and in general it does not give the multiplication structure of \(H^{*}(F;{\mathbf k})\). However, in the paper under review it is proved that for \(\operatorname {char}{\mathbf k}\) sufficiently large and a natural multiplication structure on \(Tor\), the Eilenberg-Moore isomorphism becomes one of graded algebras. The algebra of singular cochains \(C^{*}(X)\) is naturally linked to a commutative differential graded algebra \(A(X)\), inducing isomorphism in homology. The main results of the paper are the following two theorems:
Theorem A. Assume the characteristic of the field \({\mathbf k}\) is an odd prime \(p\) and consider an inclusion \(F\hookrightarrow E\) of finite \(r\)-connected CW-complexes \((r\geq 1\)) of dimension \(\leq rp\). Then the Eilenberg isomorphism \(H^{*}(F;{\mathbf k})\cong \text{Tor}^{A(B)}(A(E);{\mathbf k})\) is an isomorphism of graded algebra.
Theorem B. Let \(p\) be an odd prime. Consider the homotopy fiber \(F\) of an inclusion of finite \(r\)-connected CW-complexes of dimension \(\leq rp\). Then the cohomology algebra \(H^{*}(F;F_{p})\) is a divided powers algebra. In particular, \(p\)th powers vanish in the reduced cohomology \(\widetilde{H}^{*}(F;F_{p})\).


55R20 Spectral sequences and homology of fiber spaces in algebraic topology
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
55P62 Rational homotopy theory
57T30 Bar and cobar constructions
57T05 Hopf algebras (aspects of homology and homotopy of topological groups)
Full Text: DOI arXiv EuDML EMIS


[1] C Allday, V Puppe, Cohomological methods in transformation groups, Cambridge Studies in Advanced Mathematics 32, Cambridge University Press (1993) · Zbl 0799.55001
[2] D J Anick, Hopf algebras up to homotopy, J. Amer. Math. Soc. 2 (1989) 417 · Zbl 0681.55006
[3] L Avramov, S Halperin, Through the looking glass: a dictionary between rational homotopy theory and local algebra, Lecture Notes in Math. 1183, Springer (1986) 1 · Zbl 0588.13010
[4] H J Baues, Algebraic homotopy, Cambridge Studies in Advanced Mathematics 15, Cambridge University Press (1989) · Zbl 0688.55001
[5] H J Baues, J M Lemaire, Minimal models in homotopy theory, Math. Ann. 225 (1977) 219 · Zbl 0322.55019
[6] K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer (1994)
[7] H Cartan, S Eilenberg, Homological algebra, Princeton University Press (1956) · Zbl 0075.24305
[8] N Dupont, K Hess, Twisted tensor models for fibrations, J. Pure Appl. Algebra 91 (1994) 109 · Zbl 0789.55013
[9] S Eilenberg, J C Moore, Homology and fibrations I: Coalgebras, cotensor product and its derived functors, Comment. Math. Helv. 40 (1966) 199 · Zbl 0148.43203
[10] Y Félix, S Halperin, J C Thomas, Adams’ cobar equivalence, Trans. Amer. Math. Soc. 329 (1992) 531 · Zbl 0765.55005
[11] Y Félix, S Halperin, J C Thomas, Differential graded algebras in topology, North-Holland (1995) 829 · Zbl 0868.55016
[12] Y Félix, S Halperin, J C Thomas, Rational homotopy theory, Graduate Texts in Mathematics 205, Springer (2001)
[13] P P Grivel, Formes différentielles et suites spectrales, Ann. Inst. Fourier (Grenoble) 29 (1979) 17 · Zbl 0381.55008
[14] S Halperin, Notes on divided powers algebras,
[15] S Halperin, Lectures on minimal models, Mém. Soc. Math. France \((\)N.S.\()\) (1983) 261 · Zbl 0536.55003
[16] S Halperin, Universal enveloping algebras and loop space homology, J. Pure Appl. Algebra 83 (1992) 237 · Zbl 0769.57025
[17] D Husemoller, Fibre bundles, Graduate Texts in Mathematics 20, Springer (1994) · Zbl 0202.22903
[18] V K A M Gugenheim, J P May, On the theory and applications of differential torsion products, Memoirs of the American Mathematical Society 142, American Mathematical Society (1974) · Zbl 0292.55019
[19] S Mac Lane, Homology, Classics in Mathematics, Springer (1995) · Zbl 0818.18001
[20] M Majewski, Rational homotopical models and uniqueness, Mem. Amer. Math. Soc. 143 (2000) · Zbl 0942.55015
[21] C A McGibbon, C W Wilkerson, Loop spaces of finite complexes at large primes, Proc. Amer. Math. Soc. 96 (1986) 698 · Zbl 0594.55006
[22] B Ndombol, Algèbres de cochaînes quasi-commutatives et fibrations algébriques, J. Pure Appl. Algebra 125 (1998) 261 · Zbl 0888.55006
[23] D Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977) · Zbl 0374.57002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.