Real cohomology of the chosen torus classifying space

By [F6] and the group isomorphism [F7], the Milnor theorem's well-pointed CW-type hypothesis holds for the circle model used in each factor. The product of the $r$ free contractible circle models is free for the coordinatewise product action: if a tuple is fixed by $t=(t_1,\dots ,t_r)$, freeness in each coordinate gives $t_i=1$ for every $i$. It is contractible because a finite product of contractions is a contraction. The quotient map $E_{S^1}\to E_{S^1}\mathrm{/}{S}^1$ is open, since the saturation of an open set is a union of its group translates. Its finite product is an open surjection whose fibers are exactly the $T$-orbits, hence is a quotient map and induces $BT\cong (B{S}^1{)}^r\cong ({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}{)}^r$ by [F1]. The assertion is immediate for the empty product $r=0$.

For each $n$, [F2] makes $H_{n-1}({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }};\mathbb{Z})$ free. In [F3] with $G=\mathbb{R}$, its Ext term is therefore zero, so evaluation identifies $H^n({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }};\mathbb{R})$ with $\mathrm{Hom}(H_n({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }};\mathbb{Z}),\mathbb{R})$. The same sequence with $G=\mathbb{Z}$ identifies the integral class $u^k$ with a generator of the corresponding Hom group; naturality under $\mathbb{Z}\to \mathbb{R}$ sends it to a nonzero generator. To check multiplication is preserved, put $X={\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}$, take $\phi \in C^p(X;\mathbb{Z})$ and $\psi \in C^q(X;\mathbb{Z})$, and write $j\phi$ for postcomposition with coefficient inclusion $j:\mathbb{Z}\to \mathbb{R}$. On each singular $(p+q)$-simplex $\sigma$, the front/back formula [F8] gives $(j\phi ⌣j\psi )(\sigma )=j(\phi (\sigma [0,\dots ,p]))j(\psi (\sigma [p,\dots ,p+q]))=j((\phi ⌣\psi )(\sigma )),$ because $j$ preserves multiplication. Thus the cohomology coefficient map is a ring map and the image of $u^k$ is $u_{\mathbb{R}}^k$. Therefore $H^{\ast }({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }};\mathbb{R})=\mathbb{R}[u_{\mathbb{R}}]$, with $\mathrm{\mid }{u}_{\mathbb{R}}\mathrm{\mid }=2$, and all odd groups vanish.

Apply [F4] to the integral singular chain complex of ${\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}$ with $G=\mathbb{R}$. Each chain group is free over $\mathbb{Z}$; by [F2], every $H_{n-1}({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }};\mathbb{Z})$ is free, so the Tor term vanishes. Hence $H_n({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }};\mathbb{R})$ is $\mathbb{R}$ for even $n$ and zero for odd $n$, finite-free in each degree.

For $r\ge 1$, apply [F5] to $({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}{)}^{r-1}\times {\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}$, using the finite-free homology of the last factor established in step 1.3. Repeating this finite number of times identifies the product cohomology ring with the graded tensor product of $r$ copies of $\mathbb{R}[u_{\mathbb{R}}]$. Since each generator has even degree, the graded tensor-product signs are all positive; the result is $\mathbb{R}[u_1,\dots ,u_r]$. For $r=1$ this is step 1.2.

The symmetric algebra on the dual basis $e_1^{\ast },\dots ,e_r^{\ast }$ is the polynomial algebra on those generators. Assigning each generator degree two and sending $e_i^{\ast }$ to the factor class $u_i$ from step 2.1 defines a bijective graded-algebra homomorphism. This is the stated identification, with the chosen product presentation fixing its normalization.

If $r=0$, $ET$ and $BT$ are the one-point space and $\mathfrak{t}=0$, so both rings are $\mathbb{R}$ in degree zero. AC is used only through [F2]–[F5]; no additional arbitrary choice is made in the finite product, the naturality maps, or the generator identification.