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  "body": "\n## Statement\n\nLet $m \\ge 1$.\nGive the [[special unitary group|CZCWFPNS4N]] $SU(2m)$ the bi-invariant Riemannian metric induced by $\\langle A, B \\rangle = \\operatorname{Re} \\tr(A^* B)$.\nLet $\\Omega$ be the space of paths in $SU(2m)$ from the identity $I$ to $-I$.\n\n1. The minimal [[geodesics|NSYMQMHPDN]] from $I$ to $-I$ form a space homeomorphic to the complex [[Grassmannian|CYPA882RB9]] $G_m(\\C^{2m})$ of $m$-dimensional subspaces of $\\C^{2m}$.\n2. Every geodesic from $I$ to $-I$ that is not minimal has [[Morse index|MAX2ZYG6H9]] at least $2m + 2$, as a critical point of the energy on $\\Omega$.\n3. Hence the inclusion of $G_m(\\C^{2m})$ into $\\Omega$ induces isomorphisms $\\pi_i(G_m(\\C^{2m})) \\cong \\pi_i(\\Omega) \\cong \\pi_{i+1}(SU(2m))$ for $0 \\le i \\le 2m$.\n4. Consequently, for the stable [[unitary group|0R9R0Y1HDK]] $U = \\bigcup_n U(n)$, there are isomorphisms of [[homotopy groups|P976WYEFMB]] $\\pi_{i}(U) \\cong \\pi_{i+2}(U)$ for all $i \\ge 0$.\n\n## Remarks\n\nA geodesic from $I$ in $SU(2m)$ has the form $t \\mapsto \\exp(t X)$ with $X$ skew-Hermitian.\nIt ends at $-I$ at time $1$ exactly when the eigenvalues of $X$ are $i \\pi k_j$ with odd integers $k_j$ summing to $0$.\nThe geodesic is minimal exactly when $m$ of the $k_j$ equal $1$ and $m$ equal $-1$.\nSuch an $X$ is determined by its $m$-dimensional eigenspace for the eigenvalue $i \\pi$, which gives the Grassmannian.\nThe index of a geodesic is computed from the curvature of $SU(2m)$ through the [[Morse index theorem for geodesics|YZ4268SK8G]], and it grows with the $\\abs{k_j}$.\nMorse theory on the path space then shows that $\\Omega$ is obtained from the space of minimal geodesics by attaching cells of dimension at least $2m + 2$.\nThis gives the third statement.\nThe last statement follows from the third through the fibration $U(m) \\to V_m(\\C^{2m}) \\to G_m(\\C^{2m})$.\nIts total space, the Stiefel manifold of orthonormal $m$-frames in $\\C^{2m}$, is $2m$-connected.\nSo $\\pi_i(G_m(\\C^{2m})) \\cong \\pi_{i-1}(U(m))$ for $1 \\le i \\le 2m$, and both sides stabilize as $m$ grows.\n\nIn the same way, a chain of eight such steps through symmetric spaces gives the real periodicity $\\pi_i(O) \\cong \\pi_{i+8}(O)$ for the stable orthogonal group $O$.\nFor example, $U$ is connected and $\\pi_1(U) \\cong \\Z$ by the determinant, so $\\pi_{2k+1}(U) \\cong \\Z$ and $\\pi_{2k}(U) = 0$ for all $k \\ge 0$.\n\n## Explanations\n\n### L1 human\n\nBott periodicity says that the homotopy groups of the infinite unitary group repeat with period two.\nBott proved it by looking at geodesics in the group $SU(2m)$ joining the identity to its negative.\nThe shortest such geodesics form a Grassmannian, and all the longer ones are unstable in many directions.\nSo, up to a high dimension, the space of all paths looks like the space of shortest geodesics, and this identification gives the periodicity.\n\n### L2 human\n\nThe argument is a model application of Morse theory on path spaces.\nOne computes the space of minimal critical points of the energy and a lower bound for the indices of all other critical points.\nThen the path space and the minimal set have the same homotopy groups up to the bound.\nThe same method, applied to $Sp(n)$ and to symmetric spaces such as $SO(2n)/U(n)$, yields the eight-fold periodicity of the orthogonal group.\nBott periodicity is the foundation of topological K-theory, where it appears as $\\tilde{K}(S^{n+2}) \\cong \\tilde{K}(S^n)$.\n",
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