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...

1 Rational Homotopy Theory 4 1.1 Cohomological Eilenberg-Moore Spectral Sequences . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 k-Rational Homotopy Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Rational Homotopy Theory and E∞-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Differential Graded Lie Algebras . . . . . . . . . . ...

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...