Cup Products in Projective Spaces and Applications of Cup Products
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.
KeywordsProjective Space Commutative Diagram Quotient Space Mapping Cone Mapping Cylinder
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Bibliography for Chapter X
- H. Freudenthal, Oktaven, Ausnahme-gruppen, und Oktavengeometrie (mimeographed), Utrecht, 1951, revised ed., 1960.Google Scholar