Geometric Lefschetz number (index sum)

Let $M$ be a closed (compact, boundaryless) smooth $n$-manifold, $n\ge 1$, and let $f:M\to M$ be a smooth map all of whose fixed points are isolated (Smooth manifolds and their smooth charts, $C^r$ and smooth maps between smooth manifolds, Isolated fixed point and local fixed point index). Then $\mathrm{Fix}(f)$ is finite: it is closed, being the preimage under the continuous graph map $x\mapsto (x,f(x))$ of the diagonal, which is closed in the Hausdorff space $M\times M$ (A space is Hausdorff if and only if its diagonal is closed in the square carrying the product topology, Products of smooth manifolds have a canonical product smooth structure, Continuity of a map of topological spaces at a point and globally, Fixed points are exactly the intersections of the graph with the diagonal), and it is discrete by hypothesis, so A closed discrete subset of a compact space is finite gives finiteness. The geometric Lefschetz number of $f$ is

$I(f):={\sum }_{x\in \mathrm{Fix}(f)}{\mathrm{ind}}_x(f)\in \mathbb{Z},$

the finite sum of the local fixed point indices of Isolated fixed point and local fixed point index; for a fixed-point-free $f$ this is the empty sum $I(f)=0$. The number is defined without orienting $M$ and without any choice principle. It is not asserted here to be homotopy invariant or to equal a homology trace: those are Lefschetz-Hopf index formula and The Lefschetz number is a homotopy invariant.

• Isolatedness is a hypothesis, not a conclusion. The definition applies to every smooth self-map of a closed manifold whose fixed points are isolated, degenerate or not; for a nondegenerate fixed point the index is computed by The index of a nondegenerate fixed point is the sign of det(I-Df), but the sum itself does not require nondegeneracy. A map with non-isolated fixed points, such as the identity of a positive-dimensional closed manifold, is outside this definition; its Lefschetz number is defined algebraically in Algebraic Lefschetz number via rational homology traces, and the identity of the two notions on the overlap is Lefschetz-Hopf index formula.
• Discreteness from the subspace topology. "Isolated" means that every $x\in \mathrm{Fix}(f)$ has a neighbourhood meeting $\mathrm{Fix}(f)$ only in $x$, i.e. that $\{x\}$ is open in the subspace $\mathrm{Fix}(f)$; this is the discreteness hypothesis of A closed discrete subset of a compact space is finite, which is what makes the displayed sum finite.