Let $n \in \mathbb{Z}_{>0}$. I'm wondering about pairs $(X,V)$ where $V$ is a $2n$-dimensional vector bundle on the CW complex $X$, such that
- The ring homomorphism $\mathbb{Q}[e] \to H^*(X;\mathbb{Q})$ induced by the Euler class $e = e(V) \in H^{2n}(X)$ is an isomorphism.
- $X$ has cells only in dimension 0, $2n$, $4n$, $6n$, $\dots$.
The first property asserts a rational equivalence $X \to K(\mathbb{Z},2n)$, which is then to be factored over the map $BSO(2n) \to K(\mathbb{Z},2n)$ representing the Euler class. By an obstruction theory argument, it is not hard to prove existence of such a pair $(X,V)$, but the proof makes many choices and uses that the homotopy groups of $BSO(2n)$ are finite in all odd degrees. So I'm wondering:
How explicitly can one exhibit a pair $(X,V)$ with these two properties?
Sketch of existence, by induction on skeleta: Start with $X(2n) = S^{2n}$ and $V\vert X(2n)$ being the tangent bundle. Assume the skeleton $X(2n(k-1))$ and the restriction $V\vert X(2n(k-1))$ have been constructed, and choose a map $f': S^{2nk -1} \to X(2n(k-1))$ representing a generator of $\pi_{2nk-1}(X(2n(k-1)))\otimes \mathbb{Q} \cong \mathbb{Q}$. Then the CW complex $X' = X(2n(k-1)) \cup_{f'} D^{2nk}$ has rational cohomology ring $H^*(X';\mathbb{Q}) \cong \mathbb{Q}[x]/x^{k+1}$ for a class $x$ restricting to $e(V)$ on $X(2n(k-1))$. We are free to replace $f'$ by any non-zero multiple, and since $\pi_{2nk-1}(BO(2n))$ is a finite group, we can choose $f: S^{2nk-1} \to X(2n(k-1))$ such that the vector bundle admits an extension to $$X(2nk) = X(2n(k-1)) \cup_f D^{2nk},$$ and its euler class satisfies $e^k \neq 0 \in H^{2nn}(X(2nk)) = \mathbb{Z}$. Q.e.d.
I am curious whether one can instead just write down an explicit example without making choices. For $n=1$ we can use $\mathbb{C} P ^\infty$ and for $n=2$ we can use $\mathbb{H} P^\infty$, with their canonical bundles. For higher $n$, perhaps some other geometric construction works? (The space $X = \Omega S^{2n+1}$ has the right kind of cell structure, but I can't think of any interesting vector bundles on it.)
