A simplified index for roots (Q2493429)

Let \(X\) and \(Y\) be compact polyhedra and \(f:X \to Y\) be a map. If \(x_* \in X\), then a point \(y \in Y\) is said to be a root of \(f\) at \(x_*\) if \(f(y)=x_*\). Two roots \(y_1\) and \(y_2\) of \(F\) are said to be in the same root class if there is a path \(\alpha\) in \(Y\) joining them such that \(f \circ \alpha\) represents the trivial element of \(\pi_1(X,x_*)\). This equivalence relation partitions \(f^{-1} (x_*)\) into root classes, each of which is an isolated set in \(f^{-1}(x_*)\). Given an isolated root set \(R\) of \(f\) at \(x_*\), there is an open neighborhood \(U\) of \(R\) in \(Y\) such that \(\overline U \cap f^{-1}(x_*)=R\). The maps \[ Y \overset{j}{\longrightarrow}(Y,Y-R) \underset{e}{\longleftarrow}(\overline U , \overline U \;R) \overset{f}{\longrightarrow}(X,X-x_*) \] induce a homology homomorphism \[ f_* e^{-1}j_*: H_*(Y) \to H_*(X,X-x_*) \] called the index homomorphism or simply the index of the root set \(R\). The author studies the index in the case when \(x_*\) is a local separating point of \(X\) and shows how it can be computed.