Steenrod squares on the sphere

Question

Let $S^d$ denote the $d$-sphere. The only non-trivial cohomology groups are $H^0(S^d;\mathbb Z_2)= \mathbb Z_2$ generated by $1$ and $H^d(S^d;\mathbb Z_2)= \mathbb Z_2$ generated by the fundamental class $u$. I want to write all the Steendod squares on the sphere, that is all the group homomorphisms $$Sq^i:H^n(S^d;\mathbb Z_2)\to H^{n+i}(S^d;\mathbb Z_2)$$ satisfying some axioms.

I claim that there are only three of them:

• $Sq^0:H^0(S^d;\mathbb Z_2)\to H^0(S^d;\mathbb Z_2)$ which is the identity by the first axiom;

• $Sq^d:H^0(S^d;\mathbb Z_2)\to H^d(S^d;\mathbb Z_2)$ which sends $1$ to $u$;

• $Sq^0:H^d(S^d;\mathbb Z_2)\to H^d(S^d;\mathbb Z_2)$ which is the identity.

Is this correct?

Answer 1

So as Mariano pointed out, your second bullet is incorrect. Also, there are many squares, most of them happen to be zero.