Skip to main content

Connections between Yoneda and Pontrjagin algebras

Homotopy Theory

Part of the Lecture Notes in Mathematics book series (LNM,volume 1051)

Abstract

The relationship between Yoneda Ext algebras of local rings and the homology rings of loop spaces on simply connected CW complexes has been observed by several authors [7] [8] [11] [16]. Most of the work has been done over characteristic zero. In this paper we will use the Adams-Hilton construction [2] to understand this connection over arbitrary characteristics.

Four consequences of the resulting theory are especially noteworthy. The concept of formal spaces is generalized to non-zero characteristics. The Eilenberg-Moore spectral sequence for the homology of the loop space has E2 ≈ E as algebras for many spaces, and as algebras "up to sign" for others. We compute the Poincaré series and Pontrjagin structures for the loop space on a Λ-wedge (to be defined) of suspensions. Finally, we observe that all Ext algebras of commutative monomial k-algebras occur as the Pontrjagin rings of loop spaces.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.95
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adams, J.F., On the Non-Existence of Elements of Hopf Invariant One, Ann. of Math. 72 (1960), pp. 20–104.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Adams, J.F. and Hilton, P.J., On the Chain Algebra of a Loop Space, Comm. Math. Helv. 30 (1955), pp. 305–330.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Backelin, J., Les anneaux locaux à relations monômiales ont des séries de Poincaré-Betti rationnelles, C.R. Acad. Sci. Paris, Ser. I, t. 295 (1982), pp. 607–610.

    MathSciNet  MATH  Google Scholar 

  4. Backelin, J. and Roos, J.-E., On the Yoneda Ext Algebra of a Monomial Ring, (preprint) Reports, Department of Mathematics, Univ. of Stockholm, to appear 1983.

    Google Scholar 

  5. Baues, H. and Lemaire, J.-M., Minimal Models in Homotopy Theory, Math. Ann. 225 (1977), pp. 219–242.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Eilenberg, S. and Moore, J.C., Homology and Fibrations I, Comm. Math. Helv. 40 (1966), pp. 199–236.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Felix, Y. and Thomas, J.C., Radius of Convergence of Poincaré Series of Loop Spaces, Invent. Math. 68 (1982), pp. 257–274.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Halperin, S. and Stasheff, J., Obstructions to Homotopy Equivalences, Adv. in Math. 32, no. 3 (1979), pp. 233–279.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. James, I.M., Reduced Product Spaces, Ann. of Math. 62 (1955), pp. 170–197.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Lemaire, J.-M., Algèbres Connexes et Homologie des Espaces de Lacets, Lecture Notes in Mathematics 422, Springer-Verlag, Berlin, Heidelberg, New York.

    Google Scholar 

  11. Lemaire, J.-M., Anneaux Locaux et Especes de Lacets à Séries de Poincaré Irrationnelles, Lecture Notes in Mathematics 901, pp. 149–156, Springer-Verlag, Berlin, Heidelberg, New York.

    Google Scholar 

  12. Maclane, S., Homology (Die Grundlehren der Mathematische Wissenschaften in Einzeldarstellungen, 114), Springer-Verlag, Berlin, 1963

    CrossRef  MATH  Google Scholar 

  13. Neisendorfer, J. and Miller, T., Formal and Coformal Spaces, Ill. J. Math. 22 (1978), pp. 565–580.

    MathSciNet  MATH  Google Scholar 

  14. Porter, G.J., The Homotopy Groups of Wedges of Suspensions, Amer. J. Math. 88 (1966), pp. 655–663.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Quillen, D., Rational Homotopy Theory, Ann. of Math. 90 (1969), pp. 205–295.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. Roos, J.-E., Relations between Poincaré-Betti Series of Loop Spaces and of Local Rings, Lecture Notes in Mathematics 740, pp. 285–322, Springer-Verlag, Berlin-Heidelberg-New York.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1984 Springer-Verlag

About this paper

Cite this paper

Anick, D.J. (1984). Connections between Yoneda and Pontrjagin algebras. In: Madsen, I.H., Oliver, R.A. (eds) Algebraic Topology Aarhus 1982. Lecture Notes in Mathematics, vol 1051. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0075575

Download citation

  • DOI: https://doi.org/10.1007/BFb0075575

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12902-8

  • Online ISBN: 978-3-540-38782-4

  • eBook Packages: Springer Book Archive