Any Eilenberg-MacLane space $K(A,n)$ for abelian $A$ can be given the structure of an $H$-space by lifting the addition on $A$ to a continuous map $K(A\times A,n)=K(A,n)\times K(A,n)\to K(A,n)$.


Does somebody know an explicit way to describe this structure in the cases $K({\mathbb Z}/2{\mathbb Z},1)={\mathbb R}P^\infty$ and $K({\mathbb Z},2)={\mathbb C}P^{\infty}$?



H-space structure on infinite projective spaces
Look at $\mathbb R^\infty\setminus 0$ as the space of non-zero polynomials, which you can multiply. Pass to the quotient to construct the projective space and, from the multiplication, its $H$-space product.


The complex case is quite the same.


**NB:** Jason asks in a comment below if this is the same $H$-space structure that Hanno had in mind.
To check, we can use the fact that Hanno's is characterised by the fact that if $\mu:K(\mathbb Z\_2,1)\times K(\mathbb Z\_2,1)\to K(\mathbb Z\_2,1)$ is his product and $\alpha\in H^1(K(\mathbb Z\_2,1), \mathbb Z\_2)$ is the class represented by the identify map $K(\mathbb Z\_2,1)\to K(\mathbb Z\_2,1)$, then $\mu^\\*(\alpha)=\alpha\times 1+1\times\alpha$.
One should be able to check that this holds for the map given by multiplication of polynomials in a very small skeleton.


