$SO(n)$ is the special orthogonal group.Let $X=SO(n)\setminus\{I_n\},I_n$ is the identity matrix.Is $X$ homeomorphic to or homotopic equivalent to product of spheres?
2026-02-22 19:13:10.1771787590
Can $X=SO(n)\setminus\{I_n\}$be homeomorphic to or homotopic equivalent to product of spheres?
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Hint: $\pi_1(SO(n))=\mathbb{Z/2}.$ What happens when you remove a point?