The topological complexity sequence of a group—TC of its Milnor constructions—is weakly increasing and unbounded for infinite cohomological dimension, with controlled asymptotics for finite even-order groups.
Distributional topological complexity of groups.Preprint
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Proves dTC(Γ)=TC(Γ) for torsion-free hyperbolic and nilpotent groups, shows dTC(L^n_p)≤2p-1 and dcat(L^n_p)≤p-1 (equality in some cases), and derives counterexamples to product formulas.
Defines a new probabilistic lower-bound invariant for parametrized topological complexity and proves it matches classical behavior on Fadell-Neuwirth fibrations and sphere bundles but differs on real projective space bundles with SO structure groups.
Analog category of a finite group is essentially proportional to the order of its largest Sylow subgroup, rendering the group-order upper bound far from optimal.
citing papers explorer
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Topological complexity sequences of groups
The topological complexity sequence of a group—TC of its Milnor constructions—is weakly increasing and unbounded for infinite cohomological dimension, with controlled asymptotics for finite even-order groups.
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On distributional topological complexity of groups and manifolds
Proves dTC(Γ)=TC(Γ) for torsion-free hyperbolic and nilpotent groups, shows dTC(L^n_p)≤2p-1 and dcat(L^n_p)≤p-1 (equality in some cases), and derives counterexamples to product formulas.
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On the complexity of parametrized motion planning algorithms
Defines a new probabilistic lower-bound invariant for parametrized topological complexity and proves it matches classical behavior on Fadell-Neuwirth fibrations and sphere bundles but differs on real projective space bundles with SO structure groups.
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On the analog category of finite groups
Analog category of a finite group is essentially proportional to the order of its largest Sylow subgroup, rendering the group-order upper bound far from optimal.