Let f : E→B be a fibration of fiber F. Eilenberg and Moore have proved that there is a natural isomorphism of vector spaces H∗(F; Fp)≅TorC∗(B) (C∗(E), Fp). 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 then the algebra of singular cochains C∗(X; Fp) 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, we obtain an isomorphism of vector spaces between the algebra H∗(F; Fp) and TorA(B)(A(E), Fp) which has also a natural structure of algebra. Extending the rational case proved by Grivel–Thomas–Halperin [P P 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; Fp) is a divided powers algebra and pth powers vanish in the reduced cohomology H̃∗(F; Fp).
Menichi, Luc 1
@article{10_2140_agt_2001_1_719,
author = {Menichi, Luc},
title = {On the cohomology algebra of a fiber},
journal = {Algebraic and Geometric Topology},
pages = {719--742},
year = {2001},
volume = {1},
number = {2},
doi = {10.2140/agt.2001.1.719},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2001.1.719/}
}
Menichi, Luc. On the cohomology algebra of a fiber. Algebraic and Geometric Topology, Tome 1 (2001) no. 2, pp. 719-742. doi: 10.2140/agt.2001.1.719
[1] , , Cohomological methods in transformation groups, Cambridge Studies in Advanced Mathematics 32, Cambridge University Press (1993)
[2] , Hopf algebras up to homotopy, J. Amer. Math. Soc. 2 (1989) 417
[3] , , Through the looking glass: a dictionary between rational homotopy theory and local algebra, from: "Algebra, algebraic topology and their interactions (Stockholm, 1983)", Lecture Notes in Math. 1183, Springer (1986) 1
[4] , Algebraic homotopy, Cambridge Studies in Advanced Mathematics 15, Cambridge University Press (1989)
[5] , , Minimal models in homotopy theory, Math. Ann. 225 (1977) 219
[6] , Cohomology of groups, Graduate Texts in Mathematics 87, Springer (1994)
[7] , , Homological algebra, Princeton University Press (1956)
[8] , , Twisted tensor models for fibrations, J. Pure Appl. Algebra 91 (1994) 109
[9] , , Homology and fibrations I: Coalgebras, cotensor product and its derived functors, Comment. Math. Helv. 40 (1966) 199
[10] , , , Adams' cobar equivalence, Trans. Amer. Math. Soc. 329 (1992) 531
[11] , , , Differential graded algebras in topology, from: "Handbook of algebraic topology", North-Holland (1995) 829
[12] , , , Rational homotopy theory, Graduate Texts in Mathematics 205, Springer (2001)
[13] , Formes différentielles et suites spectrales, Ann. Inst. Fourier (Grenoble) 29 (1979) 17
[14] , Notes on divided powers algebras,
[15] , Lectures on minimal models, Mém. Soc. Math. France $($N.S.$)$ (1983) 261
[16] , Universal enveloping algebras and loop space homology, J. Pure Appl. Algebra 83 (1992) 237
[17] , Fibre bundles, Graduate Texts in Mathematics 20, Springer (1994)
[18] , , On the theory and applications of differential torsion products, Memoirs of the American Mathematical Society 142, American Mathematical Society (1974)
[19] , Homology, Classics in Mathematics, Springer (1995)
[20] , Rational homotopical models and uniqueness, Mem. Amer. Math. Soc. 143 (2000)
[21] , , Loop spaces of finite complexes at large primes, Proc. Amer. Math. Soc. 96 (1986) 698
[22] , Algèbres de cochaînes quasi-commutatives et fibrations algébriques, J. Pure Appl. Algebra 125 (1998) 261
[23] , Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977)
Cité par Sources :