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Register a new userOn the Topological Complexity of Maps
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Authors
Jamie Scott
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We define and develop a homotopy invariant notion for the topological complexity of a map $f:X to Y$, denoted TC($f$), that interacts with TC($X$) and TC($Y$) in the same way cat($f$) interacts with cat($X$) and cat($Y$). Furthermore, TC($f$) and cat($f$) satisfy the same inequalities as TC($X$) and cat($X$). We compare it to other invariants defined in the papers [15,16,17,18,20]. We apply TC($f$) to studying group homomorphisms $f:Hto G$.
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Let $mathrm{TC}_r(X)$ denote the $r$-th topological complexity of a space $X$. In many cases, the generating function $sum_{rge 1}mathrm{TC}_{r+1}(X)x^r$ is a rational function $frac{P(x)}{(1-x)^2}$ where $P(x)$ is a polynomial with $P(1)=mathrm{cat}(X)$, that is, the asymptotic growth of $mathrm{TC}_r(X)$ with respect to $r$ is $mathrm{cat}(X)$. In this paper, we introduce a lower bound $mathrm{MTC}_r(X)$ of $mathrm{TC}_r(X)$ for a rational space $X$, and estimate the growth of $mathrm{MTC}_r(X)$.
We prove the formula $TC(Gast H)=max{TC(G), TC(H), cd(Gtimes H)}$ for the topological complexity of the free product of discrete groups with cohomological dimension >2.
The Lusternik-Schnirelmann category and topological complexity are important invariants of manifolds (and more generally, topological spaces). We study the behavior of these invariants under the operation of taking the connected sum of manifolds. We give a complete answer for the LS-categoryof orientable manifolds, $cat(M# N)=max{cat M,cat N}$. For topological complexity we prove the inequality $TC (M# N)gemax{TC M,TC N}$ for simply connected manifolds.
In arXiv:1711.10132 a new approximating invariant ${mathsf{TC}}^{mathcal{D}}$ for topological complexity was introduced called $mathcal{D}$-topological complexity. In this paper, we explore more fully the properties of ${mathsf{TC}}^{mathcal{D}}$ and the connections between ${mathsf{TC}}^{mathcal{D}}$ and invariants of Lusternik-Schnirelmann type. We also introduce a new $mathsf{TC}$-type invariant $widetilde{mathsf{TC}}$ that can be used to give an upper bound for $mathsf{TC}$, $$mathsf{TC}(X)le {mathsf{TC}}^{mathcal{D}}(X) + leftlceil frac{2dim X -k}{k+1}rightrceil,$$ where $X$ is a finite dimensional simplicial complex with $k$-connected universal cover $tilde X$. The above inequality is a refinement of an estimate given by Dranishnikov.
Multiparameter persistence is a natural extension of the well-known persistent homology, which has attracted a lot of interest. However, there are major theoretical obstacles preventing the full development of this promising theory. In this paper we consider the interesting special case of multiparameter persistence in zero dimensions which can be regarded as a form of multiparameter clustering. In particular, we consider the multiparameter persistence modules of the zero-dimensional homology of filtered topological spaces when they are finitely generated. Under certain assumptions, we characterize such modules and study their decompositions. In particular we identify a natural class of representations that decompose and can be extended back to form zero-dimensional multiparameter persistence modules. Our study of this set of representations concludes that despite the restrictions, there are still infinitely many classes of indecomposables in this set.
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