If \(X\) is a finite simplicial complex and \(f: X \to X\) is a simplicial homeomorphism, show that the Lefschetz number \(\tau(f)\) equals the Euler characteristic of the set of fixed points of \(f\). In particular, \(\tau(f)\) is the number of fixed points if the fixed points are isolated. [Hint: Barycentrically subdivide \(X\) to make the fixed point set a subcomplex.]
Solution
Let \(F=\operatorname{Fix}(f)\subset X\).
1. After barycentrically subdividing \(X\), we may assume that \(F\) is a subcomplex and that every simplex carried to itself by \(f\) is fixed pointwise.
A vertex of the barycentric subdivision is the barycenter of a simplex \(\sigma\) of the original complex. A simplex in the subdivision is a strictly nested chain \[ \sigma_0\subsetneq\sigma_1\subsetneq\cdots\subsetneq\sigma_r. \] If \(f\) carries this subdivided simplex to itself, it must preserve each vertex because the dimensions of the \(\sigma_i\) are distinct. Hence it fixes the simplex pointwise. The union of such simplices is exactly the fixed-point set, so \(F\) is a subcomplex.
2. On the simplicial chain group \(C_n(X;\mathbb Q)\), the trace of \(f_*\) equals the number \(c_n(F)\) of \(n\)-simplices of \(F\).
The simplicial homeomorphism \(f\) permutes the oriented \(n\)-simplices. A simplex not carried to itself contributes zero to the diagonal of the matrix. By 1, a simplex carried to itself is fixed pointwise, hence preserves its chosen orientation and contributes \(+1\). These are exactly the \(n\)-simplices of \(F\).
3. The alternating trace on simplicial chains equals the alternating trace on homology:\[ \sum_n(-1)^n\operatorname{tr}(f_*|C_n) = \sum_n(-1)^n\operatorname{tr}(f_*|H_n). \]
For a finite chain complex of finite-dimensional vector spaces, apply additivity of trace to \[ 0\to Z_n\to C_n\to B_{n-1}\to0 \] and \[ 0\to B_n\to Z_n\to H_n\to0. \] The boundary-space trace terms cancel in the alternating sum.
4. Therefore\[ \boxed{\tau(f)=\sum_n(-1)^n c_n(F)=\chi(F).} \]
3. The alternating number of simplices of a finite simplicial complex is its Euler characteristic.
If the fixed points are isolated, the fixed subcomplex has only \(0\)-simplices, so \[ \tau(f)=\#F. \]