A variant of the topological comlexity of a map

Abstract: In this paper, we associate to two given continuous maps $f,g: X\rightarrow Z$, on a path connected space $X$, the relative topological complexity $TC^{(f, g, Z)}(X):=TC_X(X\times ZX)$ of their fiber space $X\times _ZX$. When $g=f$ we obtain a variant of the topological complexity $TC(f)$ of $f: X\longrightarrow Z$ generalizing Farber's topological complexity $TC(X)$ in the sens that $TC(X)=TC(cst{x_0})$; being $cst_{x_0}$ the constant map on $X$. Moreover, we prove that $TC(f)$ is a fiberwise homotopy equivalence invariant. When $(X,x_0)$ is a pointed space, we prove that $TC^{(f, cst_{x_0}, Z)}(X)$ interpolates $cat(X)$ and $TC(X)$ for any continuous map $f$.