For M and N closed oriented connected smooth manifolds of the same dimension, we consider the mapping space Map(M, N ; f) of continuous maps homotopic to f : M → N . We will show that the evaluation map from the space of maps to the manifold N induces a nontrivial homomorphism on the fundamental group only if the self coincidence number of f , denoted Λf,f , equals zero. Since Λf,f is equal to ...

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...

We show that the group 3i{M) of pseudoisotopy classes of diffeomorphisms of a manifold of dimension > 5 and of finite fundamental group is commensurable to an arithmetic group. As a result n0{DiffM) is a group of finite type. Let M be an «-dimensional closed smooth manifold, where n > 5, and let DiffM be the group of diffeomorphisms of M. The space DiffM (it is a topological space with the C°°-...

We prove a result on equivariant deformations of flat bundles, and as a corollary, we obtain two “splitting in a finite cover” theorems for isometric group actions on Riemannian manifolds with infinite fundamental groups, where the manifolds are either compact of Ric ≥ 0, or complete of sec ≥ 0.

The research reported in this paper explores the nature of student knowledge about group theory, and how an individual may develop an understanding of certain topics in this domain. As part of a long-term research and development project in learning and teaching undergraduate mathematics, this report is one of a series of papers on the abstract algebra component of that project. The observation...

We prove that on closed Riemannian manifolds with infinite abelian, but not cyclic, fundamental group, any isometry that is homotopic to the identity possesses infinitely many invariant geodesics. We conjecture that the result remains true if the fundamental group is infinite cyclic. We also formulate a generalization of the isometry-invariant geodesics problem, and a generalization of the cele...

In this article we find the threshold for simple connectivity of the random 2dimensional simplicial complexes Y (n, p) introduced by Linial and Meshulam [10] to be roughly p = n−1/2. One motivation for this is continuing the thread of probabilistic topology initiated by Linial and Meshulam [10], and even earlier by Erdős and Rényi [3]. (Other recent work concerning the topology of random simpli...

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.