We establish sharp upper bounds for the topological complexity TC(X) of motion planning algorithms in topological spaces X such that the fundamental group is “small”, i.e. when π1(X) is cyclic of order ≤ 3 or has cohomological dimension ≤ 2.

Let $R$ be a commutative Noetherian ring with non-zero identity and $fa$ an ideal of $R$. Let $M$ be a finite $R$--module of finite projective dimension and $N$ an arbitrary finite $R$--module. We characterize the membership of the generalized local cohomology modules $lc^{i}_{fa}(M,N)$ in certain Serre subcategories of the category of modules from upper bounds. We define and study the properti...

Let K be a 2-dimensional finite flag complex. We study the CAT(0) dimension of the ‘Bestvina-Brady group’, or ‘Artin kernel’, ΓK . We show that ΓK has CAT(0) dimension 3 unless K admits a piecewise Euclidean metric of non-positive curvature. We give an example to show that this implication cannot be reversed. Different choices of K lead to examples where the CAT(0) dimension is 3, and either (i...

A web of cohomological facts relates quantum error correction, measurement-based computation, symmetry protected topological order and contextuality. Here we extend this to computation with magic states. In computational scheme, the negativity certain quasiprobability functions is an indicator for quantumness. However, when constructing which statement applies, a marked difference arises betwee...

We study low order terms of Emerton’s spectral sequence for simply connected, simple groups. As a result, for real rank 1 groups, we show that Emerton’s method for constructing eigenvarieties is successful in cohomological dimension 1. For real rank 2 groups, we show that a slight modification of Emerton’s method allows one to construct eigenvarieties in cohomological dimension 2. Throughout th...

A group G is said to be of type FPoo if the ZG-module Z admits a projective resolution (Pi) of finite type (i.e., with each Pi finitely generated). If G is finitely presented, this is equivalent by Wall [5, 6] to the existence of an Eilenberg-Mac Lane complex K(G, 1) of finite type (i.e., with finitely many cells in every dimension). Up to now, all known torsion-free groups of type FPoo have ha...

A recent result of Bestvina and Brady [1], Theorem 8.7, shows that one of two outstanding questions has a negative answer: either there exists a group of cohomological dimension 2 and geometric dimension 3 (a counterexample to the Eilenberg-Ganea Conjecture [4]), or there exists a nonaspherical subcomplex of an aspherical 2-complex (a counterexample to the Whitehead Conjecture [11]). More preci...