Revista Matemática Iberoamericana

Homology exponents for $H$-spaces

Alain Clément and Jérôme Scherer

Full-text: Open access

Abstract

We say that a space $X$ admits a \emph{homology exponent} if there exists an exponent for the torsion subgroup of $H^*(X;\mathbb Z)$. Our main result states that if an $H$-space of finite type admits a homology exponent, then either it is, up to $2$-completion, a product of spaces of the form $B\mathbb Z/2^r$, $S^1$, $\mathbb C P^\infty$, and $K(\mathbb Z,3)$, or it has infinitely many non-trivial homotopy groups and $k$-invariants. Relying on recent advances in the theory of $H$-spaces, we then show that simply connected $H$-spaces whose mod $2$ cohomology is finitely generated as an algebra over the Steenrod algebra do not have homology exponents, except products of mod $2$ finite $H$-spaces with copies of $\mathbb C P^\infty$ and $K(\mathbb Z,3)$.

Article information

Source
Rev. Mat. Iberoamericana, Volume 24, Number 3 (2008), 963-980.

Dates
First available in Project Euclid: 9 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1228834300

Mathematical Reviews number (MathSciNet)
MR2490205

Zentralblatt MATH identifier
1160.57034

Subjects
Primary: 57T25: Homology and cohomology of H-spaces 55S45: Postnikov systems, $k$-invariants
Secondary: 55P20: Eilenberg-Mac Lane spaces 55S10: Steenrod algebra 55T10: Serre spectral sequences 55T20: Eilenberg-Moore spectral sequences [See also 57T35]

Keywords
homology exponent $H$-space loop space Steenrod algebra

Citation

Clément, Alain; Scherer, Jérôme. Homology exponents for $H$-spaces. Rev. Mat. Iberoamericana 24 (2008), no. 3, 963--980. https://projecteuclid.org/euclid.rmi/1228834300


Export citation

References

  • Bousfield, A. K.: Nice homology coalgebras. Trans. Amer. Math. Soc. 148 (1970), 473-489.
  • Bousfield, A. K.: Localization and periodicity in unstable homotopy. J. Amer. Math. Soc. 7 (1994), no. 4, 831-873.
  • Browder, W.: Torsion in $H$-spaces. Ann. of Math. (2) 74 (1961), 24-51.
  • Cartan, H.: Algèbres d'Eilenberg-MacLane et homotopie. In Séminaire Henri Cartan de l'Ecole Normale Supérieure (1954-1955), exposés 2-16. Secrétariat mathématique, 11 rue Pierre Curie, Paris, 1955.
  • Castellana, N., Crespo, J. A. and Scherer, J.: Deconstructing Hopf spaces. Invent. Math. 167 (2007), no. 1, 1-18.
  • Castellana, N., Crespo, J. A. and Scherer, J.: On the cohomology of highly connected covers of finite Hopf spaces. Adv. Math. 215 (2007), no. 1, 250-262.
  • Clément, A.: Integral cohomology of finite Postnikov towers. Ph. D. Thesis, University of Lausanne, Switzerland, 2002.
  • Clément, A.: Integral cohomology of 2-local Hopf spaces with at most two non-trivial finite homotopy groups. In An alpine anthology of homotopy theory, 87-99. Contemp. Math. 399. Amer. Math. Soc., Providence, RI, 2006.
  • Dwyer, W. G. and Wilkerson, C. W.: Spaces of null homotopic maps. Astérisque 191 (1990), no. 6, 97-108. International Conference on Homotopy Theory (Marseille-Luminy, 1988).
  • Dwyer, W. G.: The centralizer decomposition of $BG$. In Algebraic topology: new trends in localization and periodicity (Sant Feliu de Guí xols, 1994), 167-184. Progr. Math. 136. Birkhäuser, Basel, 1996.
  • Evens, L.: The cohomology ring of a finite group. Trans. Amer. Math. Soc. 101 (1961), 224-239.
  • Félix, Y., Halperin, S., Lemaire, J. M. and Thomas, J. C.: Mod $p$ loop space homology. Invent. Math. 95 (1989), no. 2, 247-262.
  • Félix, Y., Halperin, S. and Thomas, J. C.: Torsion in loop space homology. J. Reine Angew. Math. 432 (1992), 77-92.
  • Ganea, T.: A generalization of the homology and homotopy suspension. Comment. Math. Helv. 39 (1965), 295-322.
  • Grodal, J.: The transcendence degree of the mod $p$ cohomology of finite Postnikov systems. In Stable and unstable homotopy (Toronto, ON, 1996), 111-130. Fields Inst. Commun. 19. Amer. Math. Soc., Providence, RI, 1998.
  • Hubbuck, J. R. and Kane, R.: On $\pi \sb3$ of a finite $H$-space. Trans. Amer. Math. Soc. 213 (1975), 99-105.
  • Kane, R.: The homology of Hopf spaces. North-Holland Mathematical Library 40. North-Holland Publishing Co., Amsterdam, 1988.
  • Kane, R.: Personal communication. November 2006.
  • Klaus, S.: A generalization of the non-triviality theorem of Serre. Proc. Amer. Math. Soc. 130 (2002), no. 5, 1249-1256 (electronic).
  • Levi, R.: On finite groups and homotopy theory. Mem. Amer. Math. Soc. 118 (1995), no. 567.
  • Lannes, J. and Schwartz, L.: À propos de conjectures de Serre et Sullivan. Invent. Math. 83 (1986), no. 3, 593-603.
  • Miller, H.: The Sullivan conjecture on maps from classifying spaces. Ann. of Math. (2) 120 (1984), no. 1, 39-87.
  • Milnor, J. W. and Moore, J. C.: On the structure of Hopf algebras. Ann. of Math. (2) 81 (1965), 211-264.
  • McGibbon, C. A. and Neisendorfer, J. A.: On the homotopy groups of a finite-dimensional space. Comment. Math. Helv. 59 (1984), no. 2, 253-257.
  • Schwartz, L.: Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1994.
  • Selick, P.: Moore conjectures. In Algebraic topology--rational homotopy (Louvain-la-Neuve, 1986), 219-227. Lecture Notes in Math. 1318. Springer, Berlin, 1988.
  • Serre, J. P.: Cohomologie modulo $2$ des complexes d'Eilenberg-MacLane. Comment. Math. Helv. 27 (1953), 198-232.
  • Smith, L.: Lectures on the Eilenberg-Moore spectral sequence. Lecture Notes in Mathematics 134. Springer-Verlag, Berlin-New York, 1970.
  • Spanier, E. H.: Algebraic topology. Springer-Verlag, New York-Berlin, 1981. Corrected reprint.