While reading the book *Introduction to the $h$-Principle* by Y. Eliashberg and N. Mishachev, I noticed that the authors state, at the end of section 9.1.A, that the space of all symplectic structures on Euclidean space $\mathbb R^{2n}$, which they identify as $\text{GL}(2n,\mathbb R)/\text{Sp}(2n)$, * is not contractible*, whereas the space of all Euclidean structures of the aforementioned space, which can be identified with $\text{GL}(2n,\mathbb R)/\text{O}(2n)$,

*.*

**is contractible**It is easy to prove both assertions, but, * what exactly is the homotopy type of* $\text{GL}(2n,\mathbb R)/\text{Sp}(2n)$

*I have no idea of homotopy theory, so I am completely clueless.*

**?**Thanks in advance for your answers.