Abstract
In this chapter we will determine cup products in the cohomology of the real, complex, and quaternionic projective spaces. The cup products (mod 2) in real projective spaces will be used to prove the famous Borsuk—Ulam theorem. Then we will introduce the mapping cone of a continuous map, and use it to define the Hopf invariant of a map f : S 2n-1 → S n. The proof of existence of maps of Hopf invariant 1 will depend on our determination of cup products in the complex and quaternionic projective plane.
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Bibliography for Chapter X
J. F. Adams, On the nonexistence of elements of Hopf invariant one, Ann. Math., 72 (1960), 20–104.
J. Adem, The iteration of Steenrod squares in algebraic topology, Proc. Nat. Acad. Sci., 38 (1952), 720–726.
N. Bourbaki, Topologie Générale, Hermann et Cie., Paris, 1947, Chapters VI and VIII.
H. Freudenthal, Oktaven, Ausnahme-gruppen, und Oktavengeometrie (mimeographed), Utrecht, 1951, revised ed., 1960.
N. E. Steenrod, The Topology of Fibre Bundles, Princeton University Press, Princeton, 1951.
G. W. Whitehead, On the Freudenthal theorems, Ann. of Math., 57 (1953), 209–228.
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© 1980 Springer-Verlag New York Inc.
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Massey, W.S. (1980). Cup Products in Projective Spaces and Applications of Cup Products. In: Singular Homology Theory. Graduate Texts in Mathematics, vol 70. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-9231-6_10
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DOI: https://doi.org/10.1007/978-1-4684-9231-6_10
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4684-9233-0
Online ISBN: 978-1-4684-9231-6
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