Map classifying tensor products of line bundles on cohomology Let $\mu \colon \mathbb{R}P^{\infty} \times \mathbb{R} P^{\infty} \to \mathbb{R} P^{\infty}$ be the map classifying tensor products of line bundles. I want to show, that the restrictions $* \times \mathbb{R} P^{\infty} \to \mathbb{R} P^{\infty} $ and $\mathbb{R}P^{\infty} \times * \to \mathbb{R}P^{\infty} $ induce isomorphisms in singular cohomology, with coefficients in $\mathbb{F}_2$. Does anybody know how to do this?
 A: Note that $\mathbb{RP}^{\infty}$ is a $K(\mathbb{Z}_2, 1)$, so
$$[\mathbb{RP}^{\infty}, \mathbb{RP}^{\infty}] \cong [\mathbb{RP}^{\infty}, K(\mathbb{Z}_2, 1)] \cong H^1(\mathbb{RP}^{\infty}; \mathbb{Z}_2) \cong \mathbb{Z}_2.$$
Therefore every map $\mathbb{RP}^{\infty} \to \mathbb{RP}^{\infty}$ is either homotopic to a constant map, i.e. nullhomotopic, or homotopic to the identity map, in which case it is a homotopy equivalence. Since $\mathbb{RP}^{\infty}$ is a model for $BO(1)$, the classifying space of real line bundles, we also have a correspondence
$$[\mathbb{RP}^{\infty}, \mathbb{RP}^{\infty}] \cong [\mathbb{RP}^{\infty}, BO(1)] \cong \operatorname{Vect}^1(\mathbb{RP}^{\infty})$$
given by $f \mapsto f^*\gamma$. It follows that $f$ is nullhomotopic if and only if $f^*\gamma\cong\varepsilon^1$, and $f$ is a homotopy equivalence if and only if $f^*\gamma \cong \gamma$.
Now fix a basepoint $\ast \in \mathbb{RP}^{\infty}$. Then we have maps $J_1 : \mathbb{RP}^{\infty} \to \mathbb{RP}^{\infty}\times\mathbb{RP}^{\infty}$ and $J_2 : \mathbb{RP}^{\infty}\to\mathbb{RP}^{\infty}\times\mathbb{RP}^{\infty}$ given by $x \mapsto (x, \ast)$ and $y \mapsto (\ast, y)$ respectively. Note that $J_1 = (\operatorname{id}_{\mathbb{RP}^{\infty}}, c)$ and $J_2 = (c, \operatorname{id}_{\mathbb{RP}^{\infty}})$ where $c : \mathbb{RP}^{\infty}\to\mathbb{RP}^{\infty}$ is the constant map with value $\ast$. Moreover, the maps you are interested in are precisely $\mu\circ J_1$ and $\mu\circ J_2$.
As discussed in this answer, $\mu : \mathbb{RP}^{\infty}\times\mathbb{RP}^{\infty} \to \mathbb{RP}^{\infty}$ classifies the bundle $\pi_1^*\gamma\otimes\pi_2^*\gamma$ where $\pi_i : \mathbb{RP}^{\infty}\times\mathbb{RP}^{\infty} \to\mathbb{RP}^{\infty}$ is projection onto the $i^{\text{th}}$ factor, i.e. $\mu^*\gamma \cong \pi_1^*\gamma\otimes\pi_2^*\gamma$. Note that $\pi_1\circ J_1 = \pi_1\circ(\operatorname{id}_{\mathbb{RP}^{\infty}}, c)  = \operatorname{id}_{\mathbb{RP}^{\infty}}$ and $\pi_2\circ J_1 = \pi_2\circ(\operatorname{id}_{\mathbb{RP}^{\infty}}, c) = c$, so
\begin{align*}
(\mu\circ J_1)^*\gamma &\cong J_1^*\mu^*\gamma\\
&\cong J_1^*(\pi_1^*\gamma\otimes\pi_2^*\gamma)\\
&\cong J_1^*\pi_1^*\gamma\otimes J_1^*\pi_2^*\gamma\\
&\cong (\pi_1\circ J_1)^*\gamma\otimes(\pi_2\circ J_1)^*\gamma\\
&\cong \operatorname{id}_{\mathbb{RP}^{\infty}}^*\gamma\otimes c^*\gamma\\
&\cong \gamma\otimes\varepsilon^1\\
&\cong \gamma.
\end{align*}
Therefore $\mu\circ J_1 : \mathbb{RP}^{\infty}\to\mathbb{RP}^{\infty}$ is a homotopy equivalence. Likewise, $\mu\circ J_2 : \mathbb{RP}^{\infty} \to \mathbb{RP}^{\infty}$ satisfies $(\mu\circ J_2)^*\gamma \cong \gamma$ and is therefore a homotopy equivalence. In particular, both maps induce isomorphisms on cohomology.
