EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 6 June 2010. For the current production of this journal, please refer to http://www.springer.com/mathematics/geometry/journal/40062.

Reparametrizations of Continuous Paths

Reparametrizations of Continuous Paths

Ulrich Fahrenberg and Martin Raussen

A reparametrization (of a continuous path) is given by a surjective weakly increasing self-map of the unit interval. We show that the monoid of reparametrizations (with respect to compositions) can be understood via ``stop-maps'' that allow to investigate compositions and factorizations, and we compare it to the distributive lattice of countable subsets of the unit interval. The results obtained are used to analyse the space of traces in a topological space, i.e., the space of continuous paths up to reparametrization equivalence. This space is shown to be homeomorphic to the space of regular paths (without stops) up to increasing reparametrizations. Directed versions of the results are important in directed homotopy theory.

Journal of Homotopy and Related Structures, Vol. 2(2007), No. 2, pp. 93-117