spheres that are Lie groups

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spheres \PMlinkescapephrasesatisfy

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
Canonical name	SpheresThatAreLieGroups
Date of creation	2013-03-22 17:50:02
Last modified on	2013-03-22 17:50:02
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	7
Author	asteroid (17536)
Entry type	Corollary
Classification	msc 22E99
Classification	msc 57T10