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  "body": "\n## Statement\n\nLet $X$ be a compact [[smooth manifold|F45VQ47GPV]] without boundary.\nLet\n$$0 \\to \\Gamma(E_0) \\xrightarrow{d_0} \\Gamma(E_1) \\xrightarrow{d_1} \\cdots \\xrightarrow{d_{n-1}} \\Gamma(E_n) \\to 0$$\nbe an [[elliptic complex|M8GMH8GQR6]] of smooth sections of [[vector bundles|P4DZM975BV]] $E_i$ over $X$, with cohomology groups $H^i$.\nLet $f \\colon X \\to X$ be a smooth map whose fixed points are all simple, that is, $\\det(1 - df_x) \\ne 0$ at each fixed point $x$.\nFor each $i$ let $\\varphi_i \\colon f^* E_i \\to E_i$ be a smooth bundle map, and let $T_i \\colon \\Gamma(E_i) \\to \\Gamma(E_i)$ be $T_i s = \\varphi_i(s \\circ f)$.\nAssume $d_i T_i = T_{i+1} d_i$ for all $i$, so that the $T_i$ induce linear maps $H^i(T) \\colon H^i \\to H^i$.\nThen the [[Lefschetz number|ZSYCA4EGEZ]] $L(T) = \\sum_i (-1)^i \\tr H^i(T)$ satisfies\n$$L(T) = \\sum_{x \\,:\\, f(x) = x} \\frac{\\sum_i (-1)^i \\tr \\varphi_{i,x}}{\\abs{\\det(1 - df_x)}},$$\nwhere $\\varphi_{i,x} \\colon (E_i)_{f(x)} = (E_i)_x \\to (E_i)_x$ is the map on the fibre at the fixed point $x$.\n\n## Remarks\n\nThe cohomology groups are finite-dimensional because the complex is elliptic, and the fixed points are finitely many because simple fixed points are isolated and $X$ is compact.\nThe idea of the proof is to write $L(T)$ as an alternating sum of traces of operators of the form $T_i e^{-t \\Delta_i}$, where $\\Delta_i$ are the Laplacians of the complex.\nThese traces are integrals of a kernel over the graph of $f$ restricted to the diagonal, and the transversality condition $\\det(1 - df_x) \\ne 0$ localizes the integral at the fixed points as $t \\to 0$.\n\nFor the de Rham complex with $\\varphi_i = \\Lambda^i (df)^*$, the numerator at $x$ is $\\det(1 - df_x)$, so each fixed point contributes $\\sgn \\det(1 - df_x) = \\pm 1$.\nThis recovers the [[Lefschetz fixed-point theorem|G44NQXCWT9]] for maps with simple fixed points.\nFor the Dolbeault complex of a compact complex manifold and a holomorphic map $f$, it gives the holomorphic Lefschetz formula\n$$\\sum_q (-1)^q \\tr\\big(f^* \\mid H^{0,q}(X)\\big) = \\sum_{f(x) = x} \\frac{1}{\\det_\\C(1 - \\partial f_x)}.$$\n\nThe hypothesis that the fixed points are simple cannot be dropped from the formula as stated.\nFor the identity map every point is fixed, and the right side is not defined.\n\n## Explanations\n\n### L1 human\n\nThe classical Lefschetz theorem counts fixed points of a map with signs and says the count equals a trace computed from cohomology.\nAtiyah and Bott replaced the de Rham cohomology by the cohomology of any elliptic complex.\nThe global side is still a trace on cohomology.\nThe local side is a sum over the fixed points, where each point contributes a number read off from the derivative of the map at that point.\n\n### L2 human\n\nThe theorem is a fixed-point counterpart of the Atiyah–Singer index theorem: for $f = \\id$ and no fixed-point condition, $L(T)$ would be the index of the complex.\nApplied to the Dolbeault complex, it computes traces of group elements on holomorphic cohomology, which leads to the Weyl character formula for compact Lie groups acting on flag manifolds.\nApplied to the spin complex and the signature complex, it gives fixed-point formulas for the $\\hat{A}$-genus and the signature.\n",
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