A variant of the topological comlexity of a map

Abstract

In this paper, we associate to two given continuous maps $f,g:X\to Z$, on a path connected space $X$, the relative topological complexity $T{C}^{(f,g,Z)}(X):=T{C}_X(X{\times }_{Z}X)$ of their fiber space $X{\times }_{Z}X$. When $g=f$ we obtain a variant of the topological complexity $TC(f)$ of $f:X⟶Z$ generalizing Farber's topological complexity $TC(X)$ in the sens that $TC(X)=TC(cs{t}_{x_0})$; being $cs{t}_{x_0}$ 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 $T{C}^{(f,cs{t}_{x_0},Z)}(X)$ interpolates $cat(X)$ and $TC(X)$ for any continuous map $f$.

arXiv comment7 pages, submeted for publication

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