homology of $RP^\infty$ is a coalgebra. Consider (co)homology with coefficients in a field $k$ of characteristic two.
It is known that the cohomology ring $H^*(RP^\infty)$ is a polynomial algebra in one variable with coefficients in $k$.
Dualizing, we have that the homology $H_*(RP^\infty)$ inherits a structure of coalgebra with coproduct dual to the product of the polynomial algebra.
Is it possible to describe explicitly this coalgebra structure? I appreciate if anyone has any idea or a reference about it. 
 A: The dualization process is as follows : suppose $x \in H_l(\mathbb RP^\infty)$. Then by the fact that $H^l(\mathbb RP^\infty) = \hom(H_l(\mathbb RP^\infty),k)$ we get from $x$ a map $H^l(\mathbb RP^\infty)\to k$ (which is simply evaluation at $x$)
Composing with multiplication we get a map $\bigoplus_{p+q=l}H^p\otimes H^q \to k$, which is itself a sum of maps $H^p\otimes H^q\to k$
Now $H^p\otimes H^q \simeq \hom(H_p\otimes H_q, k)$ (the natural map goes $\to$, but in this situation we're fortunate enough that it is an isomorphism, so we can go $\leftarrow$)
So we get a map $\hom(H_p\otimes H_q,k)\to k$. But since everything is finite dimensional, this map is of the form $f\mapsto f(a_{p,q})$ for some $a_{p,q}\in H_p\otimes H_q$.
This tells us that we may put $\Delta (x) = \displaystyle\sum_{p+q=k}a_{p,q}$, and so we have our coalgebra structure (the co-unit being obvious)
For $\mathbb RP^\infty$, everything is one dimensional, so our task is immensely easier than in full generality : the map $H^l(\mathbb RP^\infty)\to k$ associated to each $x$ is an isomorphism and each $H^p\otimes H^q\to H^l$ is one as well. 
Up to choosing the generators correctly (look at the $k=\mathbb F_2$ case and then tensor with $k$ for instance), if $x_p$ generates $H_p(\mathbb RP^\infty)$, then $\Delta(x_l) = \displaystyle\sum_{p+q=l}x_p\otimes x_q$
(which is of course dual to how in cohomology, $x_p\cup x_q = x_{p+q}$, as $H^* \simeq k[x]$)
