A variant of the topological comlexity of a map

2018/09/26 by Rami, Youssef, Derfoufi, Younes
#55M30 #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.1809.10174

Abstract

In this paper, we associate to two given continuous maps f,g: X→ Z, on a path connected space X, the relative topological complexity TC^{(f, g, Z)}(X):=TC_X(X× _ZX) of their fiber space X× _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_x0); being cst_x0 the constant map on X. Moreover, we prove that TC(f) is a fiberwise homotopy equivalence invariant. When (X,x₀) is a pointed space, we prove that TC^(f, cst_x0, Z)(X) interpolates cat(X) and TC(X) for any continuous map f.

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