Van Kampen's Theorem and Fundamental Groupoids
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Abstract
This thesis is about Van Kampen's theorem and fundamental groupoids. Van Kampen's Theorem is a classical result in algebraic topology, which proposes a way of calculating the fundamental group of a topological spaces using the fundamental groups of certain subspaces. In this thesis we will construct the fundamental group, which intuitively counts "holes" in a topological space and is mainly used to distinguish topological spaces. Van Kampen's theorem is then proven for a arbitrary large cover using covering spaces. In the proof we glue topological spaces together and an example of this gluing is given for the Klein bottle.
Van Kampen's theorem does not work for every topological space, so we take a look at a generalization of the fundamental group, the so-called fundamental groupoid. Van Kampen's theorem can be upgraded to calculate fundamental groupoid and this theorem is proven in this thesis as well.