Classifying map of tensor product of two line bundles

Question

We know that $\mathbb{C}P^{\infty}$ is the classifying space of line bundles. Also we know that $\mathbb{C}P^{\infty}$ is an H space that is we have $$\mu: \mathbb{C}P^{\infty} \times \mathbb{C}P^{\infty} \to \mathbb{C}P^{\infty}$$ I have read that this map is the classifying map for tensor product of two line bundles but I don't understand what this means.

For example if $\xi_1$ and $\xi_2$ are two vector bundles over the space $X$ how does one use this map $\mu$ to get $\xi_1 \otimes \xi_2$? My guess is that suppose $\xi_i$ are classified by the map $f_i: X \to \mathbb{C}P^{\infty}$ . Then we have the map $g: X \to \mathbb{C}P^{\infty} \times \mathbb{C}P^{\infty}$ given by $g(x)=(f_1(x),f_2(x))$ then $\mu \circ g$ is the classifying map for the tensor product. Is my interpretation correct?

If so how does one prove this? Any hints are welcome. Thank you.

Answer 1

This isn't so much as an answer as assembling the various comments into something more cohesive.

First observe that it suffices to very this claim for the (external) tensor product of the universal line bundle $\gamma:L(\gamma)\rightarrow \mathbb{C}P^\infty$ with itself. This follows since the classifying map of any tensor product $L_1\hat\otimes L_2$ of any two lines bundles $L_i\rightarrow X_i$ will factor through this universal tensor product.

Now observe that that the H-space structure on $\mathbb{C}P^\infty\simeq K(\mathbb{Z},2)$ is unique since $H^2(\mathbb{C}P^\infty\wedge\mathbb{C}P^\infty)=0$. Therefore the homotopy class of any map $\mathbb{C}P^\infty\times \mathbb{C}P^\infty\rightarrow \mathbb{C}P^\infty$ that extends the fold $\nabla:\mathbb{C}P^\infty\vee \mathbb{C}P^\infty\rightarrow \mathbb{C}P^\infty$ along the wedge inclusion $\mathbb{C}P^\infty\vee\mathbb{C}P^\infty\hookrightarrow \mathbb{C}P^\infty\times \mathbb{C}P^\infty$ is the multplication on $\mathbb{C}P^\infty$.

If $\mu:\mathbb{C}P^\infty\times\mathbb{C}P^\infty\rightarrow \mathbb{C}P^\infty$ classifies $\gamma\hat\otimes\gamma$ then the two bundle maps $i_1:\gamma\cong \gamma\hat\otimes \epsilon\hookrightarrow\gamma\hat\otimes\gamma$, $i_2:\gamma\cong \epsilon\hat\otimes \gamma\hookrightarrow\gamma\hat\otimes\gamma$ cover the inclusions of $\mathbb{C}P^\infty$ into each factor in the product and this shows that $\mu$ extends the folding map. It follows that up to homotopy $\mu$ is the H-space multiplication.

Looking at this in a slightly different direction you can use the Segre map $\mu:\mathbb{C}P^\infty\times \mathbb{C}P^\infty\rightarrow \mathbb{C}P^\infty$ (given here https://mathoverflow.net/questions/11117/h-space-structure-on-infinite-projective-spaces) and easily lift this to a bundle map $\gamma\hat\otimes\gamma\rightarrow \gamma$ which is an isomorphism of fibres $\mathbb{C}\otimes\mathbb{C}\cong\mathbb{C}$. It follows that this map $\mu$ is classifying map for $\gamma\hat\otimes\gamma$ as well as the unique h-space multiplication on $\mathbb{C}P^\infty$.

Once you have classified the external tensor product you can then internalise it. If $L_1\rightarrow X$, $L_2\rightarrow X$ are a pair of line bundles over a space $X$ then the internal tensor product is defined by the pullback $L_1\otimes L_2=\Delta^*(L_1\hat\otimes L_2)$ where $\Delta:X\rightarrow X\times X$ is the diagonal. If $f_i:X\rightarrow \mathbb{C}P^\infty$, $i=1,2$, classifies $L_i$ then $L_1\hat\otimes L_2\cong (f_1^*\gamma)\hat\otimes(f_2^*\gamma)\cong (f_1\times f_2)^*\gamma\hat\otimes\gamma\cong (f_1\times f_2)^*\mu^*\gamma$ is classified by $\mu\circ(f_1\times f_2)$. Thus $L_1\times L_2$ is classified by the composition

$X\xrightarrow{\Delta}X\times X\xrightarrow{f_1\times f_2}\mathbb{C}P^\infty\times\mathbb{C}P^\infty\xrightarrow{\mu}\mathbb{C}P^\infty$

as your intuition suggested.