In this paper, we define the fundamental group of a topological space and explore its structure, and we proceed to prove Van-Kampen’s Theorem, a powerful result useful for calculating the fundamental groups of spaces which decompose into spaces whose fundamental groups are already known. With these tools, we show that the circle, 2-sphere, torus, and figure-8 space are topologically distinct. W...

One of the most interesting questions about a group is if its word problem can be solved and how. The word problem in the braid group is of particular interest to topologists, algebraists and geometers, and is the target of intensive current research. We look at the braid group from a topological point of view (rather than a geometrical one). The braid group is defined by the action of diffeomo...

Introduction The fundamental groupoid TT(X) of a topological space X has been known for a long time but has been regarded, usually, as of little import in comparison with the fundamental group—for example, the groupoid is described in ((3) 155) as an 'interesting curiosity'. In this paper we shall generalize the fundamental group at a point a of X, namely 7r(X,a), to the fundamental groupoid on...

We show that the topological complexity of an aspherical space X is bounded below by the cohomological dimension of the direct product A×B, whenever A and B are subgroups of π1(X) whose conjugates intersect trivially. For instance, this assumption is satisfied whenever A and B are complementary subgroups of π1(X). This gives computable lower bounds for the topological complexity of many groups ...

Anabelian geometry is a grand synthesis of number theory, algebraic geometry, algebraic topology, and group theory. Parallel to the situation in algebraic topology where to a space, one attaches a fundamental group (after fixing a basepoint), in algebraic geometry one attaches to every algebraic variety X (and a geometric point x) its étale fundamental group πét 1 (X,x). Similar to the topologi...

This paper is concerned with fixed-point free S-actions (smooth or locally linear) on orientable 4-manifolds. We show that the fundamental group plays a predominant role in the equivariant classification of such 4-manifolds. In particular, it is shown that for any finitely presented group with infinite center, there are at most finitely many distinct smooth (resp. topological) 4-manifolds which...

In this paper we prove that the fundamental group of a simplicial complex is isomorphic to the algebraic fundamental group of its incidence algebra, and we derive some applications. AMS classification : 16E40 ; 16G20 ; 06A11 ; 55Q05 Let k be a field and A be a basic and split finite dimensional k-algebra, which means that A/r = k× k× . . .× k where r is the radical of A. There exists a unique q...