Doubly connected Lorentz group?
It is said that Lorentz group $O(1,3)$ is a six-dimensional noncompact non-abelian real Lie group that is not connected. The four connected components are not simply connected, but rather doubly connected.
What is the definition of the doubly connected? Does it mean that $\pi_2(O(1,3))=0$?
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$\begingroup$In this context, "doubly connected" means that $\pi_1$ is a group of order $2$. $SO^{+}(1, 3)$ has universal cover the indefinite spin group $\text{Spin}(1, 3)$, which turns out to be isomorphic to $SL_2(\mathbb{C})$.
This isn't great terminology because, as you say, it can be misread as meaning $2$-connected, which means that (the space is path-connected and) both $\pi_1$ and $\pi_2$ vanish. Some people also use "$n$-tuply connected" to mean that there are $n$ path components; I would avoid that terminology too.
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