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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.
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The third author noticed in his 1992 PhD Thesis [Sim92] that every regular tree language of infinite trees is in a class $Game (D_n({bfSigma}^0_2))$ for some natural number $ngeq 1$, where $Game$ is the game quantifier. We first give a detailed exposition of this result. Next, using an embedding of the Wadge hierarchy of non self-dual Borel subsets of the Cantor space $2^omega$ into the class ${bfDelta}^1_2$, and the notions of Wadge degree and Veblen function, we argue that this upper bound on the topological complexity of regular tree languages is much better than the usual ${bfDelta}^1_2$.
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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 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 $f:Hto G$.
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.
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