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 ⩽ r ⩽ n. We shall prove this Poincaré duality theorem for general homology theories.
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References
J. F. Adams [8]
M. F.Atiyah[15]
E. H. Spanier [79, 80]
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© 2002 Springer-Verlag Berlin Heidelberg
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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
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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
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