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  "body": "\n## Statement\n\nLet $G$ be a connected [[Lie group|7JBRGCVJRB]].\nThen:\n\n1. $G$ has a [[maximal compact subgroup|T2EB8R9HEV]], and each compact subgroup of $G$ is contained in a maximal compact subgroup;\n2. any two maximal compact subgroups of $G$ are conjugate in $G$;\n3. each maximal compact subgroup $K$ is connected, and $G$ is homeomorphic to $K \\times \\R^n$ for some $n \\ge 0$.\n\n## Remarks\n\nFor semisimple groups with finite center, the result follows from the [[Cartan decomposition|A5JB9JQKR0]] $G = K \\exp(\\mathfrak{p})$ and from Cartan's fixed point theorem.\nThat theorem says that a compact group acting by isometries on the nonpositively curved space $G/K$ has a fixed point.\nSo each compact subgroup lies in the stabilizer of a point, a conjugate of $K$.\nThe general case is due to Malcev and Iwasawa.\n\nExample: $G = \\mathrm{GL}_n^+(\\R)$, the invertible real matrices of positive determinant.\nA maximal compact subgroup is $\\mathrm{SO}(n)$, and the polar decomposition gives a homeomorphism of $G$ with $\\mathrm{SO}(n) \\times \\R^{n(n+1)/2}$, the second factor being the symmetric matrices.\n\nThe conclusions extend to Lie groups with finitely many connected components, except that $K$ then has one component for each component of $G$.\nFor groups with infinitely many components, maximal compact subgroups may fail to exist.\n\n## Explanations\n\n### L1 human\n\nEvery connected Lie group contains a largest bounded subgroup, unique up to conjugation.\nThe whole group is this bounded subgroup times a flat space with no holes.\nSo all the topology of the group, for example its loops and its holes, lives in the compact part.\n\n### L2 human\n\nThe theorem reduces the homotopy type of a connected Lie group to that of a compact Lie group.\nFor example $\\mathrm{SL}_2(\\R)$ has the homotopy type of a circle, and $\\mathrm{SL}_n(\\C)$ that of $\\mathrm{SU}(n)$, which is simply connected.\nIt also says that the classifying space of $G$ is that of $K$, which is why characteristic classes of $G$-bundles only depend on $K$.\nUniqueness up to conjugacy makes $K$ an invariant of $G$, used for example to define the symmetric space $G/K$.\n",
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  "msc": [
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  "name": "Cartan–Iwasawa–Malcev theorem",
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