If the fundamental group of a symplrctic Lie group be infinite cyclic, why it should has a unique connected double cover?
2026-03-31 19:14:23.1774984463
A question about double cover of Lie group
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When $X$ is a "well-connected" space (path-connected, locally path connected, semilocally simply connected), equivalence classes of connected $n$-fold covers correspond to conjugcay classes of subgroups of $\pi_1(X)$ which are of index $n$.
The hypothesis above is satisfied by any connected manifold. So connected $n$-fold covers of the symplectic group correspond to subgroups of $\mathbb{Z}$ of index $n$. Since there is only one such subgroup, namely $2\mathbb{Z}$, there is (up to equivalence) only one connected double cover.