Steenrod's interpretation of the Hopf invariant.

Question

Where can I read about Steenrod's interpretation of the Hopf invariant?

Is there any reference?

Answer 1

I suppose you mean the definition in terms of Steenrod squares, which are a type of 'cohomology operation'.

The idea is basically, you have a map $f:S^{2n-1}\to S^n$. Interpret $S^{2n-1}$ as the boundary of $D^{2n}$, and form the space $S^n \cup_f D^{2n}$, which is the 2n-disk glued to the n-sphere along its boundary via f.

Thinking cellularly, it is obvious that the (co)homology of this space is zero except in degree $n$ and degree $2n$, where it is $\mathbb{Z}$.

Since cohomology is a ring, given a cohomology class $[c]$ in degree $n$ we can form a new class $[c]^2$ of degree $2n$. If we pick generators $x$ for $H^n$ and $y$ for $H^{2n}$, we have $x^2 = h(f) * y$ for some number $h(f)$, depending on $f$ but only up to homotopy. This number is the Hopf invariant.

Steenrod showed that the squaring operation you get from the ring structure ($H^n\to H^{2n}:: [c] \to [c]^2$) is actually part of a much more elaborate set of operators $H^*\to H^{*+n}::[c]\to Sq^n([x])$ (there are different operations, indexed by $n$, called 'Steenrod squares' because if you apply them to an $n$-dimensional cohomology class you get the usual square with the cup-product). Notice the situation is actually more complicated than I am describing: the Steenrod squares are for coefficients $\mathbb{Z}/2\mathbb{Z}$ (exercise: when is $H^n\to H^{2n}:[c]\to[c]^2$ a group homomorphism, cf the discussion around 'Hopf invariant 1' problems), and there are other operations.

You can read more in "Cohomology Operations" by Steenrod and Epstein, or "Cohomology Operations: with applications to Homotopy Theory" by Mosher and Tangora.