The focus of this exposition is the fundamental group of a topological space. The paper begins by defining paths and homotopy, and proceeds to construct the fundamental group. Later, attention turns to the consequences of the construction, in particular to the fundamental group of S1 and Van Kampen’s Theorem.

These notes, from a first course in algebraic topology, introduce the fundamental group and the fundamental groupoid of a topological space and use them to classify covering spaces.

Closed, connected, nonorientable, topological 4-manifolds with infinite cyclic fundamental group are classified. The classification is an extension of results of Freedman and Quinn and of Kreck. The stable classification of such 4-manifolds is also obtained. In principle, it is a consequence of Freedman’s work on topological 4manifolds that one can classify topological 4-manifolds with “good” f...

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.

We deene a new invariant for symbolic Z 2-actions, the projective fundamental group. This invariant is the limit of an inverse system of groups, each of which is the fundamental group of a space associated with the Z 2-action. The limit group measures a kind of long-distance order that is manifested along loops in the plane, and roughly speaking bears the same relation to the mixing properties ...

0BQ6 Contents 1. Introduction 1 2. Schemesétale over a point 1 3. Galois categories 2 4. Functors and homomorphisms 9 5. Finité etale morphisms 11 6. Fundamental groups 14 7. Topological invariance of the fundamental group 15 8. Finité etale covers of proper schemes 17 9. Local connectedness 19 10. Fundamental groups of normal schemes 24 11. Group actions and integral closure 26 12.

In the study of Fuchsian groups, it is a nontrivial problem to determine a set of generators. Using a dynamical approach we construct for any cocompact arithmetic Fuchsian group a fundamental region in SL2(R) from which we determine a set of small generators.

In this paper, a certain new connectedness of L-fuzzy subsets inL-topological spaces is introduced and studied by means of preclosed sets. Itpreserves some fundamental properties of connected set in general topology.Especially the famous K. Fan’s Theorem holds.