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