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  "body": "\n## Statement\n\nLet $M$ be a closed oriented smooth manifold of dimension $4k$.\nLet a compact Lie group $G$ act smoothly on $M$, and let $g \\in G$ act by an orientation-preserving diffeomorphism.\nThe [[intersection form|WK8GKRKAVD]] on $H^{2k}(M; \\R)$ is $g$-invariant, so $H^{2k}(M; \\R) = H^+ \\oplus H^-$ with $g$-invariant subspaces on which the form is positive and negative definite.\nThe $g$-signature is $\\operatorname{sign}(g, M) = \\operatorname{tr}(g \\mid H^+) - \\operatorname{tr}(g \\mid H^-)$.\nLet $M^g$ be the fixed point set of $g$, a disjoint union of closed submanifolds $F$.\nThen\n$$\\operatorname{sign}(g, M) = \\sum_{F} \\langle L(g, F), [F] \\rangle.$$\nHere each $L(g, F) \\in H^*(F; \\C)$ is a characteristic class.\nIt is determined by the tangent bundle of $F$, the normal bundle of $F$ in $M$ and the action of $g$ on that normal bundle.\nIn particular:\n\n1. If $g$ has no fixed points, then $\\operatorname{sign}(g, M) = 0$.\n2. If $g$ is the identity, the formula is the [[Hirzebruch signature theorem|TEG1ZX1AGJ]].\n\n## Remarks\n\nThe theorem is the [[equivariant index theorem|T866256Y1W]] for the [[signature operator|4NCYA7GME1]], whose equivariant index evaluated at $g$ is $\\operatorname{sign}(g, M)$; the topological side localizes to the fixed point set.\nFor an isolated fixed point the contribution is an explicit product of cotangents of half the rotation angles of $g$ on the tangent space, up to a power of $i$.\nExample: let a finite group $\\Gamma$ act freely and preserving orientation on $M$.\nThen every $g \\ne 1$ has $\\operatorname{sign}(g, M) = 0$, and averaging the characters gives $\\sigma(M) = \\abs{\\Gamma} \\, \\sigma(M / \\Gamma)$, where $\\sigma$ is the [[signature|07EZA80NEH]] of the intersection form.\n\n## Explanations\n\n### L1 human\n\nA symmetry of a manifold acts on its middle cohomology and preserves the intersection form.\nThe $g$-signature measures this action by a trace.\nThe theorem says that this trace can be computed only by looking near the points that the symmetry fixes.\n\n### L2 human\n\nThe $G$-signature theorem of Atiyah and Singer is a basic tool in the study of group actions on manifolds.\nIt gives obstructions to actions with prescribed fixed point data, through the cotangent expressions that appear at isolated fixed points.\nIt is a model for later fixed point formulas from the equivariant index theorem.\n",
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