General linear group over reals: diffeomorphism

Question

What does the general linear group over reals look like; i.e., what is it diffeomorphic to? I mean some other manifold I can picture if any?

Answer 1

Any connected Lie group $G$ is diffeomorphic to its maximal compact subgroup $K$ times a Euclidean space. In this case, you can use Gram-Schmidt, or more precisely QR factorization, to show that $GL_n(\mathbb{R})$ is diffeomorphic to $O(n) \times \mathbb{R}^{\frac{n(n+1)}{2}}$. In turn, $O(n)$ is diffeomorphic (although not isomorphic) to $SO(n) \times \mathbb{Z}_2$. So the problem reduces to understanding what $SO(n)$ looks like as a manifold. Here is the answer for small values of $n$:

• $SO(1)$ is a point.
• $SO(2)$ is the circle $S^1$.
• $SO(3)$ is the real projective space $\mathbb{RP}^3$. It is double covered by $Spin(3) \cong SU(2)$, which is the $3$-sphere $S^3$.
• $SO(4)$ is a $6$-manifold with fundamental group $\mathbb{Z}_2$; it has no name more familiar than $SO(4)$. It is double covered by $Spin(4) \cong SU(2) \times SU(2)$, which is a product of two $3$-spheres $S^3 \times S^3$.

Beyond these cases $SO(n)$ has no name more familiar than $SO(n)$ even up to taking covering spaces. But there are other questions you could ask from here. For example, the special orthogonal groups fit into a sequence of fiber bundles

$$SO(n-1) \to SO(n) \to S^{n-1}$$

coming from the action of $SO(n)$ on the unit sphere in $\mathbb{R}^n$, and so loosely speaking $SO(n)$ is like an "iterated twisted product" of the spheres $S^{n-1}, S^{n-2}, \dots$ (in particular it has the same dimension as the product of these spheres). When $n = 3$ this fiber bundle is a variant of the Hopf fibration.

On the other hand, any connected Lie group is rationally homotopy equivalent to a product of odd spheres, and it is possible to name which odd spheres occur in the case of $SO(n)$. I've already said which ones occur up to $n = 4$; the pattern is hard to guess because it depends on the parity of $n$, but for example, $SO(5)$ is rationally homotopy equivalent to $S^3 \times S^7$.