# spheres that are Lie groups

Theorem - The only spheres (http://planetmath.org/Sphere) that are Lie groups are $S^{0}$, $S^{1}$ and $S^{3}$.

$\,$

Proof: $\;$ It is known that $S^{0}$, $S^{1}$ and $S^{3}$ have a Lie group .

On the other , we have seen in the parent entry (http://planetmath.org/CohomologyOfCompactConnectedLieGroups) that the cohomology groups (http://planetmath.org/DeRhamCohomology) of a compact connected Lie group $G$ satisfy

 $H^{1}(G;\mathbb{R})=0\;\,\Longrightarrow\;\,H^{3}(G;\mathbb{R})\neq 0$

The result then follows from the fact that the of spheres satisfy $H^{1}(S^{n};\mathbb{R})=0$ and $H^{3}(S^{n};\mathbb{R})=0$ for $n\neq 1,3$. $\square$

Title spheres that are Lie groups SpheresThatAreLieGroups 2013-03-22 17:50:02 2013-03-22 17:50:02 asteroid (17536) asteroid (17536) 7 asteroid (17536) Corollary msc 22E99 msc 57T10