TITLE: Why does one think to Steenrod squares and powers?
QUESTION [42 upvotes]: I'm studying Steenrod operations from Hatcher's book. Like homology, one can use them only knowing the axioms, without caring for the actual construction. But while there are plenty of intuitive reasons to introduce homology, I cannot find any for the Steenrod operations. I can follow the steps in the proofs given by Hatcher, but I don't understand why one introduces all these spaces like $\Lambda X$, $\Gamma X$ and so on (in Hatcher's notation, I don't know if it's universal). Does anyone know how to get an intuitive grasp of what's going on?

REPLY [38 votes]: Here's how I explain Steenrod squares to geometers.  First, if $X$ is a manifold of dimension $d$ then one can produce classes in $H^n(X)$ by proper maps $f: V \to X$ where $V$ is a manifold of dimension $d-n$ through many possible formalisms - eg. intersection theory (the value on a transverse $n$-cycle is the count of intersection points), or using the fundamental class in locally finite homology and duality, or Thom classes, or as the pushforward $ f_*(1)$  where $1$ is the unit class in $H^0(V)$.   Taking this last approach, suppose $f$ is an immersion and thus has a normal bundle $\nu$.  If $x = f_*(1) \in H^n(X)$ then $Sq^i(x) = f_*(w_i(\nu))$.  This is essentially the Wu formula.
That is, if cohomology classes are represented by submanifolds, and for example cup product reflects intersection data, then Steenrod squares remember normal bundle data.