Tamanoi equation for orbifold Euler characteristics revisited (Q6633150)

Let \(\chi(X)\) denote the Euler characteristic of a topological space \(X\). If \(X\) has an action of a finite group \(G\), one can define the orbifold Euler characteristic \(\chi^{\mathrm{orb}}(X,G)\) and the higher order orbifold Euler characteristics \(\chi^{(k)}(X,G)\), \(k\geq 0\), of \(X\) that take into account the group action. Then \(\chi^{(0)}(X,G)=\chi(X/G)\) is the Euler characteristic of the quotient \(X/G\) and \(\chi^{(1)}(X,G)=\chi^{\mathrm{orb}}(X,G)\).\N\NThere are generalizations of the higher order orbifold Euler characteristics \(\chi^{(k)}\) that depend on \(X\), \(G\) and an arbitrary finitely generated group \(A\). The orbifold Euler characteristic of the \(G\)-space \(X\), corresponding to the finitely generated group \(A\) (or the \(A\)-Euler characteristic), is defined by the equation \N\[\N\chi^{(A)}(X,G)=\frac{1}{\vert G\vert}\sum_{\varphi\in {\mathrm{Hom}}(A,G)}\chi(X^{\langle \varphi(A)\rangle}). \N\]\NHere the summation is over the set of homomorphisms \(\varphi\colon A\to G\), and \(X^{\langle\varphi(A)\rangle}\) denotes the set of points of \(X\) that are fixed by all the elements in the image of \(\varphi\). By choosing \(A\) to be the free abelian group \({\mathbb{Z}}^k\), one obtains the characteristic \(\chi^{(k)}\).\N\NThe Macdonald equation states that, for a sufficiently nice topological space \(X\), \N\[\N1+\sum_{k=1}^\infty \chi(S^kX)t^k=(1-t)^{-\chi(X)},\N\]\Nwhere \(S_k\) denotes the permutation group of \(k\) elements, and \(S^kX=X^k/S_k\) is the \(k\)th symmetric power of \(X\). The Tamanoi equation is a Macdonald-type equation for the orbifold Euler characteristic \(\chi^{\mathrm{orb}}\) and for its higher order analogs \(\chi^{(k)}\). The author reduces the proof of the Tamanoi equation to the case of actions of finite groups on the one-point space, i.e., to the \(G\)-space \(G/G\). He generalizes the statements used for the proof to the characteristics \(\chi^{(A)}\), and shows that an analog of the Bryan-Fulman equation, i.e., the Tamanoi equation for the trivial group \(G\), holds for the \(A\)-Euler characteristic with an arbitrary finitely generated group \(A\). However, an analog of the Tamanoi equation itself does not hold in general.