Algebraic Topology

In this paper we discuss a specific follow-up result from the work about the 'theory of p-nomial triangles'. We want to present some key developments. They involve the representation of twelve connected p-nomial triangles in three dimensional space in the form of a Magen David star pyramid. We construct the representation via coefficient [boundary] vectors, before we show that the established Magen David star pyramid is homotopic to the dodecagonal -.

This undergraduate honours thesis examines the Thurston norm by an application of normal surface theory, shedding light on the topological signicance and further potential applications arising from this approach.
In particular, provide a means to explore whether the polytope structure of the unit ball corresponds to notable topological results, and whether there is recognisable regularity to the norm ball on a combinatorial level. This thesis presents an algorithm for computing the Thurston norm unit ball via the approach of normal surface theory, and hence the Thurston norm of any homology class.

The present paper is a note on the relative tensor degree of finite groups. This notion generalizes the tensor degree, introduced recently in literature, and allows us to adapt the concept of relative commutativity degree through the notion of nonabelian tensor square. We show two inequalities, which correlate the relative tensor degree with the relative commutativity degree of finite groups.

The fixed point properties and abelianization of arithmetic subgroups $\Gamma$ of $\operatorname{SL}_n(D)$ and its elementary subgroup $\operatorname{E}_n(D)$ are well understood except in the degenerate case of lower rank, i.e. $n=2$ and $\Gamma = \operatorname{SL}_2(\mathcal{O})$ with $\mathcal{O}$ an order in a division algebra $D$ with only a finite number of units. In this setting we determine property (FA) for $\operatorname{E}_2(\mathcal{O})$ and its subgroups of finite index. Along we construct a generic and computable exact sequence describing $\operatorname{E}_2(\mathcal{O})^{ab}$. We also consider $\operatorname{E}_n(R)$ for $n \geq 3$ and $R$ an arbitrary finitely generated ring for which we give a short algebraic proof of the existence of a global fixed point when acting on $(n-2)$-dimensional CAT(0) cell complexes. The latter is called property (FA)$_{n-2}$. Thenceforth, we investigate applications in integral representation theory. We obtain a group and ring theoretic...

Using the notion of discrete Morse function introduced by R. Forman for finite cw-complexes, we generalize it to the infinite 2-dimensional case in order to get the corresponding version of the well-known discrete Morse inequalities on a non-compact triangulated 2-manifold without boundary and with finite homology. We also extend them for the more general case of a non-compact triangulated 2-pseudo-manifold with a finite number of critical simplices and finite homology.

where μA (resp. μB) is the multiplication of A (resp. B). We deduce that the ”natural” tensor product μA⊗B = μA ⊗ μB provides A⊗ B with a Lie-admissible algebra structure. In [2] we have defined special classes of Lie-admissible algebras with relations of definition determined by an action of the subgroups Gi of the 3-degree symmetric group Σ3. We obtain quadratic operads, denoted by Gi−Ass and in this family we find operads of Lie-admissible, associative, Vinberg and pre-Lie algebras. For these operads we have proved that for every P-algebra A and P -algebra B the tensor product A⊗B is a P-algebra. This is not true for general nonassociative algebras and, in this sense, the Gi-associative algebras are the most regular kind of nonassociative algebras. For example if P is the operad of Leibniz algebras or of the nonassociative algebras associated to Poisson algebras [?], then the tensor product of a P-algebra and P -algebra is not a P-algebra. So we introduce a quadratic operad, deno...

The aim of this paper We create requirements for a graph cover to have the homotopy lifting property of topological space covers, or A-Homotopy lifting property. Also, it defines the homotopy cover for a graph  and gives a definition of the covering graph found in [1], and develops the path-lifting property.