Universality and Examples in the Context of Functorial Semi-Norms

Let F: C → Vect be a functor from a category C to vector spaces over a normed field. A functorial semi-norm on F is a factorization of F over the forgetful functor snVect → Vect, where snVect denotes the corresponding category of semi-normed vector spaces. Functorial semi-norms, in particular the ℓ¹-semi-norm, on singular homology were introduced by Gromov in his study of simplicial volume of manifolds. The latter is a homotopy invariant that, roughly speaking, measures the complexity of the fundamental class of an oriented closed connected manifold. In the present thesis, we investigate three aspects of functorial semi-norms: Universal functorial semi-norms (joint work with Clara Löh). For a fixed functor, we define a relation among all its functorial semi-norms, whose minimal elements we call universal. We then prove certain existence results of such universal functorial semi-norms. Inflexibility from a computational perspective. Crowley and Löh established a bidirectional correspondence between functorial semi-norms on singular homology and so-called inflexible manifolds. The construction of such manifolds is based on the construction and study of certain differential graded algebras, which are purely algebraic objects. As such they are very amenable to computations, not only by humans but also via computer programs. We present fragments of a software that facilitates such computations, some new examples that we found in this way, and two results about algorithmic decidability. Excisive approximation of ℓ¹-homology. It is a well-known fact, that ℓ¹-homology does not satisfy the excision axiom in the sense of Eilenberg and Steenrod. On the other hand, the fact that singular homology satisfies excision is already visible at the level of the singular chain complex functor, namely the latter is excisive in the sense of Goodwillie calculus. The latter, however, also provides the framework for constructing a universal (or best) excisive approximation to a given functor. We apply this theory to the ℓ¹-chain complex functor and show that its excisive approximation vanishes. In appendices, we include a proof of the fact that the singular chain complex functor is excisive in the sense of Goodwillie calculus, and we relate this abstract form of excision to the classical one in the form of Mayer-Vietoris sequences

Non-associative Gröbner bases, finitely-presented Lie rings and the Engel condition, II

AbstractWe give an algorithm for constructing a basis and a multiplication table of a finite-dimensional finitely-presented Lie ring. Secondly, we give relations that are equivalent to the n-Engel condition, and only have to be checked for the elements of a basis of a Lie ring. We apply this to construct the freest t-generator Lie rings that satisfy the n-Engel condition, for (t,n)=(2,3),(3,3),(4,3),(2,4)