On the role of 1-LC and semi 1-LC properties in determining the fundamental group of a one point union of spaces

Abstract

Concerning the fundamental group of spaces written as a union of two topological spaces, the result by Seifert and van Kampen is well known and frequently used. Over the years there have been various theorems published on the topic of fundamental groups of the unions of spaces. A portion of those theorems deal with spaces whose intersection is a one point set. In 1954 Griffiths' result was published stating the following theorem. Theorem 0.2 If one of the spaces X1 or X2 is 1-LC at x, and both X1 and X2 are closed in X and satisfy the first axiom of countability, then the free product of the fundamental groups of X1 and X2 is isomorphic to the fundamental group of X, where X is the union of X1 and X2 and the intersection of X1 and X2 is a one point set {x}. The goal of this paper is to show that the 1-LC property in Griffiths' result cannot be generalized to semi 1-LC. Spanier indicated this problem in one of his homework exercises. This result is not, however, contained in Griffiths' paper or Spanier's book. To provide the necessary proof two spaces are constructed with the following properties: 1. X is semi 1-LC (but not 1-LC) 2. the fundamental group of X at x_0 is trivial 3. the fundamental group of the one point union of X with itself is not trivial.