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

##### Date

2006-05-15##### Author

Moore, Emilia

##### Type of Degree

Thesis##### Department

Mathematics and Statistics##### Metadata

Show full item record##### 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.