In the study of topology, we are often interested in understanding and classifying the internal structure of topological spaces. Algebraic topology is the application of abstract algebra to topology in order to further identify the structure of topological spaces by developing a correspondence between topological spaces and certain groups called homotopy groups. In this paper, we will examine t...

Given a finite connected bipartite graph B = (X, Y ) we consider the simplicial complexes of complete subgraphs of the square B of B and of its induced subgraphs B[X] and B[Y ]. We prove that these three complexes have isomorphic fundamental groups. Among other applications, we conclude that the fundamental group of the complex of complete subgraphs of a graph G is isomorphic to that of the cli...

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.

Let M be a connected smooth G-manifold, where G is a connected compact Lie group. In this paper, we first study the relation between π1 (M) and π1 (M/G). Then we particularly focus on the case when M is a connected Hamiltonian G-manifold with an equivariant moment map φ. In [13], for compact M , we proved that π1 (M) = π1 (M/G) = π1 (Ma) for all a ∈ image(φ), where Ma is the symplectic quotient...

Hecke proved analytically that when h > 2 or when h = 2 CDS (m/q), q E Z, q > 3, then B(h) = (7: Im I > 0, / Re 7 1 < h/2, [ 7 j > 1) is a fundamental region for the group G(h) = (SA , T), where S,+ : 7 --f 7 + X and T: 7 ---t -l/~. He also showed that B(A) fails to be a fundamental region for all other A > 0 by proving that G(X) is not discontinuous. We give an elementary proof of these facts ...

We address the problem of computing the fundamental group of a symplectic S-manifold for non-Hamiltonian actions on compact manifolds, and for Hamiltonian actions on non-compact manifolds with a proper moment map. We generalize known results for compact manifolds equipped with a Hamiltonian S-action. Several examples are presented to illustrate our main results.

One of the main tools for the study of the category of finite dimensional modules over a basic algebra, over an algebraically closed field k is its presentation as quiver and relations. This theory is mainly due to P. Gabriel (see for example [GRo]). More precisely, it has been proved that for all finite dimensional and basic algebras over an algebraically closed field k, there exists a unique ...

We prove a local, unipotent, analog of Kedlaya’s theorem for the pro-p part of the fundamental group of integral affine schemes in characteristic p.

We analyze irreducible plane sextics whose fundamental group factors to D14. We produce explicit equations for all curves and show that, in the simplest case of the set of singularities 3A6, the group is D14 × Z3.

The associahedron is an object that has been well studied and has numerous applications, particularly in the theory of operads, the study of non-crossing partitions, lattice theory and more recently in the study of cluster algebras. We approach the associahedron from the point of view of discrete homotopy theory, that is we consider 5-cycles in the 1-skeleton of the associahedron to be combinat...