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  "body": "\n## Definition\n\nLet $X$ be a finite [[CW complex|5W95QFHXPR]], or more generally a space homotopy equivalent to one, and let $f \\colon X \\to X$ be a [[continuous map|0VGFHTM75Y]].\nFor each $k \\ge 0$, let $f_{\\ast,k} \\colon H_k(X; \\Q) \\to H_k(X; \\Q)$ be the induced linear map on [[homology with rational coefficients|XT7VWNTKGB]], a finite-dimensional vector space.\nThe *Lefschetz number* of $f$ is\n$$L(f) = \\sum_{k \\ge 0} (-1)^k \\tr\\big(f_{\\ast,k}\\big),$$\nwhere $\\tr$ is the [[trace|0KZDNGA561]].\nThe sum is finite: if $X$ is homotopy equivalent to a finite CW complex of dimension $d$, then $H_k(X; \\Q) = 0$ for $k \u003e d$.\n\n## Remarks\n\nHomotopic maps induce the same maps on homology, so $L(f)$ depends only on the homotopy class of $f$.\nFor the identity map, $L(\\id_X) = \\sum_k (-1)^k \\dim H_k(X; \\Q) = \\chi(X)$, the [[Euler characteristic|RNDDWDAFJ5]].\nOne may use $H_k(X; \\Z)$ modulo torsion instead of rational homology and get the same number.\n\nExample: let $X$ be connected and let $f$ be constant.\nThen $f_{\\ast,0}$ is the identity of $H_0(X; \\Q) \\cong \\Q$ and $f_{\\ast,k} = 0$ for $k \u003e 0$, so $L(f) = 1$.\n\nExample: the antipodal map $a$ of $S^n$ acts on $H_n(S^n; \\Q)$ by its degree $(-1)^{n+1}$, so $L(a) = 1 + (-1)^n (-1)^{n+1} = 0$, in line with the fact that $a$ has no fixed point.\n\nNon-example: for an infinite wedge of circles, $H_1(X; \\Q)$ is infinite-dimensional, so the trace in the formula is not defined; the finiteness hypothesis on $X$ fails.\n",
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