Skip to main content

Hodge Theory for Riemannian Solenoids

Part of the Springer Optimization and Its Applications book series (SOIA,volume 52)

Abstract

A measured solenoid is a compact laminated space endowed with a transversal measure. The De Rham L 2-cohomology of the solenoid is defined by using differential forms which are smooth in the leafwise directions and L 2 in the transversal direction. We develop the theory of harmonic forms for Riemannian measured solenoids, and prove that this computes the De Rham L 2-cohomology of the solenoid.This implies in particular a Poincaré duality result.

Keywords

  • Solenoids
  • Harmonic forms
  • Cohomology
  • Hodge theory

Mathematics SubjectClassification (2000): Primary 58A14; Secondary 57R30, 58A12

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-1-4614-0055-4_39
  • Chapter length: 25 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   219.00
Price excludes VAT (USA)
  • ISBN: 978-1-4614-0055-4
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   279.99
Price excludes VAT (USA)
Hardcover Book
USD   279.99
Price excludes VAT (USA)

References

  1. Dodziuk, J.: Sobolev spaces of differential forms and De Rham-Hodge isomorphism. J. Diff. Geom. 16, 63–73 (1981)

    MathSciNet  MATH  Google Scholar 

  2. Heitsch, J.L.: A cohomology of foliated manifolds. Comment. Math. Helvetici 50, 197–218 (1975)

    MathSciNet  MATH  CrossRef  Google Scholar 

  3. Macias, E.: Continuous cohomology of linear foliations on T 2. Rediconti di Matematica Serie VII (Roma) 11, 523–528 (1991)

    Google Scholar 

  4. Massey, W.S.: Singular Homology Theory. Graduate Texts in Mathematics 70, Springer-Verlag (1980)

    Google Scholar 

  5. Moore, C., Schochet, C.: Global analysis on foliated spaces. Mathematical Sciences Research Institute Publications 9, Springer-Verlag (1988)

    Google Scholar 

  6. Muñoz, V., Pérez-Marco, R.: Ergodic solenoids and generalized currents. Preprint.

    Google Scholar 

  7. Muñoz, V., Pérez-Marco, R.: Schwartzman cycles and ergodic solenoids. Preprint.

    Google Scholar 

  8. Muñoz, V., Pérez-Marco, R.: Ergodic solenoidal homology: Realization theorem. Preprint.

    Google Scholar 

  9. Muñoz, V., Pérez-Marco, R.: Ergodic solenoidal homology II: Density of ergodic solenoids. Aust. J. Math. Anal. Appl. 6, no. 1, Article 11, 1–8 (2009)

    Google Scholar 

  10. Reinhart, B.L.: Harmonic integrals on almost product manifolds. Trans. Amer. Math. Soc. 88, 243–276 (1958)

    MathSciNet  MATH  CrossRef  Google Scholar 

  11. Ruelle, D., Sullivan, D.: Currents, flows and diffeomorphisms. Topology 14, 319–327 (1975)

    MathSciNet  MATH  CrossRef  Google Scholar 

  12. Wells, R.O.: Differential analysis on complex manifolds. GTM 65, Second Edition, Springer-Verlag (1979)

    Google Scholar 

Download references

Acknowledgement

Partially supported through grant MEC (Spain) MTM2007-63582.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Vicente Muñoz .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2011 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Muñoz, V., Marco, R.P. (2011). Hodge Theory for Riemannian Solenoids. In: Rassias, T., Brzdek, J. (eds) Functional Equations in Mathematical Analysis. Springer Optimization and Its Applications(), vol 52. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0055-4_39

Download citation