Fixed point indices of equivariant maps and Möbius inversion (Q1088252)

Let G be a compact Lie group, and C(G) the set of conjugacy classes of closed subgroups of G. Let \(f: X\to X\) be a G-map of a G-ENR X. If the fixed point set of f is compact, the fixed point index \(\iota (f^{(K)})\) is defined for any (K)\(\in C(G)\), where \(f^{(K)}=f| X^{(K)}: X^{(K)}\to X^{(K)}\), \(X^{(K)}\) is the subspace of points whose isotropy subgroups contain a subgroup conjugate to K. We then show that there exists a unique family of integers \(\{a_{(H)}(f)| \in C(G)\}\) such that \(\iota (f^{(K)})=\sum _{(K)\leq (H)}\chi (G/H)a_{(H)}(f)\), where \(\chi\) ( ) denotes the Euler characteristic. Using the Möbius inversion formula, from the equality above we obtain \(\sum _{(H)\leq (K)}\mu (H),(K))\iota (f^{(K)})\equiv 0 mod \chi (G/H)\) for any fixed (H)\(\in C(G)\), where \(\mu\) (, ) is the Möbius function on some subset of C(G). If \(G=Z_ n\) is a cyclic group, then the congruences for some \(Z_ n\)-maps reduce to the congruences for iterated maps obtained by \textit{A. Dold} [Invent. Math. 74, 419-435 (1983; Zbl 0583.55001)]. We also obtain similar congruences as above in which the Möbius function \(\mu\) (, ) is replaced by an Euler function \(\phi\) (, ).