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Spaces with high topological complexity

Published online by Cambridge University Press:  24 July 2014

Aleksandra Franc  and
Petar Pavešić
Aleksandra Franc
Affiliation:
Faculty of Computer and Information Science, University of Ljubljana, Tržaška 25, 1000 Ljubljana, Slovenia, [email protected]
Petar Pavešić
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 21, 1000 Ljubljana, Slovenia, [email protected]
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Abstract

By a formula of Farber, the topological complexity TC(X) of a (p − 1)-connected m-dimensional CW-complex X is bounded above by (2m + 1)/p + 1. We show that the same result holds for the monoidal topological complexity TCM(X). In a previous paper we introduced various lower bounds for TCM(X), such as the nilpotency of the ring H*(X × X, Δ(X)), and the weak and stable (monoidal) topological complexity wTCM(X) and σTCM(X). In general, the difference between these upper and lower bounds can be arbitrarily large. In this paper we investigate spaces with topological complexity close to the maximal value given by Farber's formula. We show that in these cases the gap between the lower and upper bounds is narrow and TC(X) often coincides with the lower bounds.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 144 , Issue 4 , August 2014 , pp. 761 - 773
Copyright
Copyright © Royal Society of Edinburgh 2014 

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