We introduce fibrewise Whitehead- and fibrewise Ganea definitions of monoidal topological complexity. We then define several lower bounds for the topological complexity, which improve on the standard lower bound in terms of nilpotency of the cohomology ring. The relationships between these lower bounds are studied.Comment: 19 pages, 1 figur

Abstract. By a formula of Farber [7, Theorem 5.2] the topological complexity TC(X) of a (p − 1)-connected, m-dimensional CW-complex X is bounded above by (2m + 1)/p + 1. There are also various lower estimates for TC(X) such as the nilpotency of the ring H * (X × X, ∆(X)), and the weak and stable topological compexity wTC(X) and σTC(X) (see [10]). In general the difference between these upper and lower bounds can be arbitrarily large. In this paper we investigate spaces whose topological complexity is close to the maximal value given by Farber's formula and show that in these cases the gap between the lower and upper bounds is narrow and that TC(X) often coincides with the lower bounds.

Given a filtration function on a finite simplicial complex, stability theorem of persistent homology states that the corresponding barcode is continuous with respect to changes in the filtration function. However, due to the discrete setting of simplicial complexes, the critical simplices terminating matched bars cannot change continuously for arbitrary perturbations of filtration functions. In this paper we provide a sufficient condition for rigidity of a terminal simplex, i.e., a condition on ε > 0 implying that the terminal simplex of a homology class or a bar in persistent homology remains constant through ε-perturbations of filtration function. The condition for a homology class or a bar in dimension n depends on the barcodes in dimensions n and n + 1.

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Given a filtration function on a finite simplicial complex, stability theorem of persistent homology states that the corresponding barcode is continuous with respect to changes in the filtration function. However, due to the discrete setting of simplicial complexes, the simplices terminating matched bars cannot change continuously for arbitrary perturbations of filtration functions. In this paper we provide a sufficient condition for rigidity of a terminal simplex, i.e., a condition on $$\varepsilon >0$$ ε > 0 implying that the terminal simplex of a homology class or a bar in persistent homology remains constant through $$\varepsilon $$ ε -perturbations of filtration function. The condition for a homology class or a bar in dimension n depends only on the barcodes in dimensions n and $$n+1$$ n + 1 .