Products of complex projective spaces span rational oriented bordism

The rank-$r$ Thom space is $(r-1)$-connected. By [F1], $T_r$ is a based CW space whose non-basepoint cells have dimension at least $r$; applying [F2] to the inclusion of the basepoint gives ${\pi }_i(T_r)=0$ for $i\le r-1$. With $c=r$ and $i=n+r$ the inequalities of [F2] hold exactly when $r\ge n+2$, and in that range the actual rational Hurewicz map ${\pi }_{n+r}(T_r)\otimes \mathbb{Q}\to H_{n+r}(T_r;\mathbb{Q})$ is an isomorphism; since $n+r>0$, reduced and ordinary homology agree here and no injectivity at the upper endpoint $2r-1$ is used.

Pass to the stable colimit. By the Pontryagin-Thom isomorphism [F3], ${\mathrm{\Omega }}_n^{SO}\cong {\mathrm{colim}}_r{\pi }_{n+r}(T_r)$. Tensoring with $\mathbb{Q}$ commutes with the sequential colimit: presenting the colimit as the cokernel of the shift map on the direct sum, exactness of $-\otimes \mathbb{Q}$ and its compatibility with direct sums [F3] identify $({\mathrm{colim}}_r{\pi }_{n+r}(T_r))\otimes \mathbb{Q}$ with ${\mathrm{colim}}_r({\pi }_{n+r}(T_r)\otimes \mathbb{Q})$, and the tail $r\ge n+2$ is cofinal. Naturality of the Hurewicz map with respect to suspension followed by the structure map ${\alpha }_r$ makes the isomorphisms of step 1.1 a map of sequential systems, so ${\mathrm{\Omega }}_n^{SO}\otimes \mathbb{Q}\cong {\mathrm{colim}}_{r\ge n+2}{H}_{n+r}(T_r;\mathbb{Q})$.

The homology transition maps are isomorphisms. Let $i_r:B_r\to B_{r+1}$ classify ${\epsilon }_{+}^1\oplus {\gamma }_r^{+}$, the base map of the structure map. The normalized Thom classes and the ordered suspension normalization give the commuting Thom square of [F4], in which the bottom map is identified with $i_r^{\ast }$ after the reduced cohomology suspension: $H^n(B_{r+1};\mathbb{Q})\cong {\tilde{H}}^{n+r+1}(T_{r+1};\mathbb{Q})\to {\tilde{H}}^{n+r}(T_r;\mathbb{Q})\cong H^n(B_r;\mathbb{Q})$. Since the degree-$n$ part of the polynomial presentation [F4] uses only the stable Pontryagin generators $p_j$ with $4j\le n$, which occur in every rank $s>n$, naturality and stability give that $i_r^{\ast }:H^n(B_{r+1};\mathbb{Q})\to H^n(B_r;\mathbb{Q})$ is an isomorphism for every $r\ge n+1$, in particular throughout the tail.

Duality and dimension count. By [F5] the evaluation map is a natural isomorphism $H^q(Y;\mathbb{Q})\cong {\mathrm{Hom}}_{\mathbb{Q}}(H_q(Y;\mathbb{Q}),\mathbb{Q})$ for every space. The Thom cohomology and the polynomial presentation show that ${\tilde{H}}^{n+r}(T_r;\mathbb{Q})$ is finite-dimensional; hence $H_{n+r}(T_r;\mathbb{Q})$ is finite-dimensional of the same dimension, since an infinite-dimensional vector space would have an infinite-dimensional dual using a basis and its coordinate functionals. Naturality of evaluation identifies the dual of the homology transition in step 2.1 with the cohomology transition of step 3.1, which is an isomorphism; therefore every homology transition is an isomorphism and the colimit of step 2.1 is any one of its tail groups, giving ${\dim }_{\mathbb{Q}}({\mathrm{\Omega }}_n^{SO}\otimes \mathbb{Q})={\dim }_{\mathbb{Q}}{H}^n(B_r;\mathbb{Q})$ for $r\ge n+2$.

Identifying the dimension. If $n=4k$, the degree-$4k$ monomials in the universal Pontryagin classes are exactly $p_1^{a_1}{p}_2^{a_2}\cdots$ with ${\sum }_j{a}_j=k$, indexed by the partitions of $k$; hence the dimension is $p(k)$, the number of partitions of $k$. If $4\nmid n$ no monomial has degree $n$, so the dimension is zero. For $k=0$ there is one degree-zero monomial, and [F5] identifies ${\mathrm{\Omega }}_0^{SO}\otimes \mathbb{Q}\cong \mathbb{Q}$ with the positively oriented point.

Conclusion. For $k\ge 1$, [F5] supplies $p(k)$ linearly independent classes $[P_J]$ in degree $4k$, and step 5.1 shows the dimension is exactly $p(k)$; hence they form a basis, and the Pontryagin-number matrix on this basis is invertible by the triangularity lemma [F5]. The product theorem for bordism makes $x_j\mapsto [{\mathbb{C}\mathbb{P}}^{2j}]$ a graded ring map from the polynomial algebra on the classes $[{\mathbb{C}\mathbb{P}}^{2j}]$ to ${\mathrm{\Omega }}_{\ast }^{SO}\otimes \mathbb{Q}$; on each degree $4k$ it carries the monomials to the basis just proved and both sides vanish in degrees not divisible by four, so it is an isomorphism of graded rings, giving uniqueness of the polynomial expression. The empty product in degree zero is the point.