Talk:Hopf invariant

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

Thanks
Ooh, thanks for adding the references to those papers!


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

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 $$S^{2n+1}$$ 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 $$\alpha^{\cup n+1} = h(f)\beta$$?