Map classifying tensor products of line bundles on cohomology

Question

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?

Answer 1

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.