the Lefschetz fixed-point theorem
/ LEF-shets /
When you continuously deform a space onto itself, must some point stay put? Sometimes yes, sometimes no — a rotation of the circle moves every point, but a rotation of the disk fixes the center. The Lefschetz fixed-point theorem answers this with a single integer computed from homology: if that integer is nonzero, a fixed point is guaranteed. It is a vast generalization of Brouwer's theorem, turning a question about maps into a question about traces.
Let f: X -> X be a continuous self-map of a compact space with finitely generated rational homology (a finite CW complex, say). The map induces linear maps f_*: H_k(X; Q) -> H_k(X; Q) on each rational homology vector space. The Lefschetz number is the alternating sum of their traces, L(f) = sum over k of (-1)^k trace(f_* on H_k(X; Q)). The theorem states: if L(f) is not zero, then f has at least one fixed point. Equivalently, contrapositively, a fixed-point-free map must have Lefschetz number zero. The proof compares the graph of f with the diagonal in X cross X and computes their intersection number two ways.
Special cases recover famous results. If f is homotopic to the identity, then f_* is the identity on each H_k, the traces are the Betti numbers, and L(f) becomes the alternating sum b_0 - b_1 + b_2 - ... = chi(X), the Euler characteristic. So a self-map homotopic to the identity has a fixed point whenever chi(X) is nonzero — which is exactly why every continuous self-map of an even sphere, or of any space of nonzero Euler characteristic, behaves rigidly, and why a nowhere-zero vector field cannot exist on such a space (the hairy ball theorem in disguise). Taking X a disk and noting it is contractible recovers Brouwer's fixed-point theorem.
The honest caveats run both directions. The theorem is one-directional: L(f) nonzero forces a fixed point, but L(f) = 0 does NOT mean there is no fixed point — it only means the fixed points (if any) cancel in the count. A rotation of the circle with no fixed points has L = 0, consistent with the theorem, but so does the identity on the circle, which fixes everything; L = 0 simply gives no information. Also, the theorem needs rational (or real) coefficients and a finiteness hypothesis; over the integers traces are not generally defined, and for infinite complexes the Lefschetz number need not even exist.
Any continuous self-map f of the 2-sphere homotopic to the identity has L(f) = chi(S^2) = 1 - 0 + 1 = 2, which is nonzero, so f must fix a point. By contrast the antipodal map a(x) = -x acts as -1 on H_2(S^2) = Z, giving L(a) = 1 + (-1) = 0 — and indeed the antipodal map has no fixed point, consistent with L = 0.
On S^2 the identity has L = 2 (forced fixed point) but the antipodal map has L = 0 (none).
The implication runs one way only: nonzero Lefschetz number forces a fixed point, but L(f) = 0 says nothing — fixed points may exist and merely cancel. The theorem needs rational coefficients and a finiteness hypothesis; integral traces are not generally available.