Algebraic Lefschetz number via rational homology traces

Assume the Axiom of Choice (The Axiom of Choice). Let $M$ be a closed smooth $n$-manifold and let $f:M\to M$ be continuous. Its Lefschetz number is $L(f):={\sum }_{i=0}^n(-1{)}^i\mathrm{tr}(f_{\ast }:H_i(M;\mathbb{Q})\to H_i(M;\mathbb{Q}))\in \mathbb{Q},$ the alternating sum of the traces of the induced rational homology endomorphisms (The singular chain complex and singular homology, The basis-independent trace of an endomorphism of a finite-dimensional vector space, The rationals as equivalence classes of pairs of integers). The sum is finite and every trace is of a finite-dimensional operator because ${\dim }_{\mathbb{Q}}{H}_i(M;\mathbb{Q})<\mathrm{\infty }$ and $H_i(M;\mathbb{Q})=0$ for $i>n$ (Finiteness and additivity of the Euler characteristic, clause (i), which is where the Axiom of Choice enters, through the existence of an excellent Morse function (Every compact smooth manifold admits an excellent Morse function)). When a finite CW model whose cellular chains compute $H_{\ast }(M;\mathbb{Q})$ is available (The handle chain complex computes singular homology, Cellular homology computes singular homology), this number agrees with the published finite-CW Lefschetz number Lefschetz number of a finite CW self-map of the induced self-map transported along the homotopy equivalence (Homotopy equivalences induce isomorphisms on singular homology); the comparison is a consistency statement, not part of the definition. Homotopic maps have the same $L$ because they induce the same maps on homology (Homotopic maps induce the same map on singular homology). No orientation, no smoothness of $f$ and no field other than $\mathbb{Q}$ is used.

• Why the sum stops at $n$. A closed $n$-manifold has $H_i(M;\mathbb{Q})=0$ for $i>n$, by the finiteness proposition cited above, which is proved from the existence of an excellent Morse function on the double of $M$ and the in-run handle chain complex. This is the only place where the Axiom of Choice is used in the definition; the trace is a basis-independent invariant of an endomorphism of a finite-dimensional $\mathbb{Q}$-vector space (The basis-independent trace of an endomorphism of a finite-dimensional vector space).

• Value field. Each rational homology trace, and hence $L(f)$, is an integer. Choose the finite CW model from the Morse handle presentation and A handle decomposition gives a relative CW complex, and transport $f$ using a homotopy inverse. Cellular approximation (Cellular approximation for maps of CW pairs) gives a homotopic cellular map with integral matrices on the finite free cellular chains (Relative homology of consecutive CW skeleta, Cellular maps induce cellular chain maps). Its cycles are finite free (A submodule of a free module of finite rank over a PID is free of no larger rank), so integral homology is finitely generated. Quotienting torsion gives a finite free lattice (Every finitely generated torsion-free module over a PID is free). Clearing denominators in rational cycles shows that this lattice spans rational homology; clearing denominators in a rational bounding chain shows that its kernel before quotienting is exactly torsion. Thus rational homology is the rationalization of the lattice, and the induced map has an integral lattice matrix and integer trace. Cellular comparison and homology conjugation give the same trace for $f_{\ast }$. Alternatively Hopf trace formula computes $L(f)$ as the alternating integral cellular-chain trace.

• Not the same as the geometric number. This definition applies to every continuous self-map, including maps with non-isolated fixed points such as the identity; the equality with the finite index sum $I(f)={\sum }_x{\mathrm{ind}}_x(f)$ for smooth self-maps of closed smooth $n$-manifolds with $n\ge 1$ and isolated fixed points is Lefschetz-Hopf index formula, and the corresponding geometric number is Geometric Lefschetz number (index sum).