Skip to main content

Orientation and Duality

  • Chapter
  • 3039 Accesses

Part of the book series: Classics in Mathematics ((CLASSICS,volume 212))

Abstract

We have seen how to construct homology and cohomology theories E*, E* out of a spectrum E, and in the last chapter we showed how to construct various products connecting E* and E* if E is a ring spectrum. In many cases, however, we can establish a much stronger connection between E*(X) and E*(X) for suitable spaces X. One of the early discoveries of algebraic topology was that if M is a closed n-dimensional orientable manifold, then H r (M; ℤ) ≌ Hn-r(M; ℤ) for all r, 0 ⩽ rn. We shall prove this Poincaré duality theorem for general homology theories.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.99
Price excludes VAT (USA)
  • 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. J. F. Adams [8]

    Google Scholar 

  2. M. F.Atiyah[15]

    Google Scholar 

  3. E. H. Spanier [79, 80]

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Switzer, R.M. (2002). Orientation and Duality. In: Algebraic Topology — Homotopy and Homology. Classics in Mathematics, vol 212. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61923-6_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-61923-6_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42750-6

  • Online ISBN: 978-3-642-61923-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics