---
tag: 6PWN57BW45
kind: definition
name: Lefschetz number
msc: [55M20]
defines: ZSYCA4EGEZ
uses: [5W95QFHXPR, 0VGFHTM75Y, XT7VWNTKGB, 0KZDNGA561, RNDDWDAFJ5]
sources:
  - url: https://en.wikipedia.org/wiki/Lefschetz_fixed-point_theorem
    locator: Formal statement
---

## Definition

Let $X$ 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]].
For 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.
The *Lefschetz number* of $f$ is
$$L(f) = \sum_{k \ge 0} (-1)^k \tr\big(f_{\ast,k}\big),$$
where $\tr$ is the [[trace|0KZDNGA561]].
The sum is finite: if $X$ is homotopy equivalent to a finite CW complex of dimension $d$, then $H_k(X; \Q) = 0$ for $k > d$.

## Remarks

Homotopic maps induce the same maps on homology, so $L(f)$ depends only on the homotopy class of $f$.
For the identity map, $L(\id_X) = \sum_k (-1)^k \dim H_k(X; \Q) = \chi(X)$, the [[Euler characteristic|RNDDWDAFJ5]].
One may use $H_k(X; \Z)$ modulo torsion instead of rational homology and get the same number.

Example: let $X$ be connected and let $f$ be constant.
Then $f_{\ast,0}$ is the identity of $H_0(X; \Q) \cong \Q$ and $f_{\ast,k} = 0$ for $k > 0$, so $L(f) = 1$.

Example: 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.

Non-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.