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  "body": "\n## Description\n\nLet $M$ be a smooth manifold and $S^1 = \\R / \\Z$ the [[circle|060W5CBZP6]].\nThe *free loop space* of $M$ is\n$$L M = C^\\infty(S^1, M),$$\nthe set of all smooth loops in $M$, with no condition on a base point.\nSince $S^1$ is compact, it is a manifold of smooth mappings, which is a Fréchet manifold.\nIts tangent space at a loop $\\gamma$ is the space of smooth vector fields along $\\gamma$, that is, [[sections|WWX7H5W1RP]] of $\\gamma^* TM$.\n\n## Remarks\n\n\nThe circle acts on $L M$ by rotating the parameter, $(\\theta \\cdot \\gamma)(t) = \\gamma(t + \\theta)$, and the fixed points of this action are the constant loops, a copy of $M$.\nEvaluation at $0$ is a map $L M \\to M$, $\\gamma \\mapsto \\gamma(0)$, whose fibre over $x$ is the smooth based [[loop space|2JFF5F3EGF]] $\\Omega_x M$.\nFor $M = S^1$, a loop has a degree in $\\Z$, and the connected components of $L S^1$ correspond to the integers.\n\nIn topology one also uses the space $\\mathrm{Map}(S^1, X)$ of continuous loops in a space $X$ with the [[compact-open topology|RG15F0TWPW]].\nFor a manifold $M$, the inclusion of smooth loops into continuous loops is a homotopy equivalence.\n\n\n## Explanations\n\n### L1 human\n\nA point of the free loop space is a whole closed curve in $M$, which may sit anywhere and need not pass through a chosen point.\nMoving a little in this space means wiggling the curve.\nConstant curves, which sit still at one point, form a copy of $M$ inside the loop space.\nThe free loop space is where one looks for closed geodesics, which are the critical points of the energy of a loop.\n",
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