$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\PSp{PSp}\DeclareMathOperator\USp{USp}\DeclareMathOperator\BSp{BSp}\DeclareMathOperator\BUSp{BUSp}\DeclareMathOperator\BPSp{BPSp}$Let $\USp(n,\mathbb{C})$ and $\Sp(n,\mathbb{C})$ be the compact symplectic group and the symplectic group, respectively. This is,
$$
\Sp(n,\mathbb{C})=\left\{A\in M(2n,\mathbb{C}): A^{tr}J\_{2n}A=J\_{2n}\right\}
$$
where $M(n,\mathbb{C})$ is the set of $n\times n$ matrices with entries in $\mathbb{C}$,
$$
J\_{2n}=\begin{pmatrix}
0 & I\_{n}\\
-I\_{n} & 0\\
\end{pmatrix},
$$
and $\USp(n,\mathbb{C})$ is isomorphic to $\Sp(n,\mathbb{C})\cap \mathrm{U}(2n,\mathbb{C})$.
Let $\PSp(n,\mathbb{C})$ be the projective symplectic group, i.e. $\Sp(n,\mathbb{C})/\{\pm I\_{2n}\}$.


It can be shown that the inclusion $\USp(n,\mathbb{C}) \to \Sp(n,\mathbb{C})$ is a homotopy equivalence.


[Hatcher, AT, pg 271 and 381] says that the quaternionic Grassmannian $G\_{m}(\mathbb{H}^{\infty})$ is a model for $\BUSp(n,\mathbb{C})$, and that the integral cohomology $H^{\*}(G\_{m}(\mathbb{H}^{\infty}))\cong \mathbb{Z}[\alpha\_{1},\alpha\_{2},\dots,\alpha\_{m}]$ with $|\alpha\_{i}|=4i$.


This result makes you think that $\BSp(n,\mathbb{C})$ has a CW structure with cells only in degrees $4i$. I would like to find such structure for $\BSp(n,\mathbb{C})$ and also, know whether this structure induces one of the same kind in $\BPSp(n,\mathbb{C})$.


The closest thing I know about their CW structures is that $G\_{m}(\mathbb{H}^{\infty})$ has a CW structure with each $G\_{m}(\mathbb{H}^{k})$ a finite subcomplex, [Hatcher, VB and Kthy, pg 31-34].


Thank you for your ideas.



CW structure for $\mathrm{BSp}(n,\mathbb{C})$ and $\mathrm{BPSp}(n,\mathbb{C})$ in degrees $4i$
I'll assume that by $BG$ you mean ``any space of the form $E/G$, where $E$ is a contractible space with free $G$-action''. (The alternative would be to define $BG$ as the geometric realisation of a specific simplicial space, but then $BU(n)$ would not have even cells.) But then you can take $G=Sp(n,\mathbb{C})=\text{Aut}\_{\mathbb{H}}(\mathbb{H}^n)$ and $E=\text{Inj}\_\mathbb{H}(\mathbb{H}^n,\mathbb{H}^\infty)$ and we get $BSp(n,\mathbb{C})=G^{\mathbb{H}}\_n(\mathbb{H}^\infty)$. This has a CW structure based on quaternionic Schubert cells (which have real dimension divisible by $4$) just as in the real and complex cases. We can also describe $G^{\mathbb{H}}\_n(\mathbb{H}^\infty)$ as the space of isometric $\mathbb{H}$-linear embeddings $\mathbb{H}^n\to\mathbb{H}^\infty$ mod the action of isometric $\mathbb{H}$-linear automorphisms of $\mathbb{H}^n$, so $G^{\mathbb{H}}\_n(\mathbb{H}^\infty)$ qualifies as $BUSp(n,\mathbb{C})$ as well as $BSp(n,\mathbb{C})$.


There is an inclusion $SO(3)\simeq PSU(2)\to PSp(1,\mathbb{C})$, which is a homotopy equivalence and so gives a homotopy equivalence of classifying spaces. It is known that $H^\*(BSO(3);\mathbb{Z}/2)\simeq\mathbb{Z}/2[w\_2,w\_3]$ with $|w\_2|=2$ and $|w\_3|=3$. This implies that any CW complex homotopy equivalent to $PSp(1,\mathbb{C})$ must have odd-dimensional cells.


