The Lusternik-Schnirelmann category of the Lie groups $Sp(n)$. Since $Sp(1) = S^3$, $\mathrm{cat}(Sp(1)) = 1$. In the 1960s, P. Schweitzer proved that $\mathrm{cat}(Sp(2)) = 3$. Based on this, a folklore conjecture emerged that in general $\mathrm{cat}(Sp(n)) = 2n+1$. In 2001, it was proved that $\mathrm{cat}(Sp(3)) = 5$, so maybe it's true?
Jeff Strom
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