Talk:Hopf invariant

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I noticed this article was missing and gave a little exposition of the subject. If anyone would like to add relevant cross references, that'd be good. If you'd like any clarifications, let me know. - Thomas

[edit] Thanks

Ooh, thanks for adding the references to those papers!

No problem. Ryan Reich 15:50, 27 March 2006 (UTC)

[edit] Generalisations

According to the article on the Hopf bundle, the construction can also be generalised to maps

f \colon S^{2n+1} \to \mathbb{C}P^n

by viewing S2n + 1 as a set of n-tuples (z_1,\ldots,z_{n+1}) \in \mathbb{C}^{n+1} which are never all zero and mapping them to [z_1:\ldots:z_{n+1}] \in \mathbb{C}P^n. We can view \mathbb{C}P^n as a cell complex with one k-cell in dimension 2k for k=1,\ldots,n, and we can attach a 2n + 2-cell via f to obtain a new space X. This operation does not change the lower cohomology, and we get again one generator \alpha \in H^2(\mathbb{C}P^n) and one generator \beta \in H^{2n+2}(X).

Would it make sense to define an invariant h(f) such that \alpha^{\cup n+1} = h(f)\beta?