The thesis is divided into two part::;. Firstly, in Chapter 1 we
prove Union Theorems for both double groupoids end groupoids. Brown-Higgins
define the homotopy double groupoid p(x,Y,Z) of' a triple of spaces when
each loop in Z is contractible in Y. They prove the Union Theorem: if
U - fX I 1.'3 a family of zubset~ of X- t A A€A - whose interiors cover X , then
U p(X,Y,z) ,2 Y Y Y V€J4c-~ p(X, Y ,z) is a ccequaliser diagram under certain connectivity conditions which are conditions on 8-fold intersections for both and In § 1 we prove that the above diagram is a coequaliser under minimal connectivity condl tions, namely conditions on 4-fold inter.sections for 7tO Mel
conditions on 3-fold intersections for P1 • We also glve examples to show
that the connectivity assumptions are best possible. In § 2 , Vie gcneralise
§ 1 by constructinG a homotopy double groupoid p(q,p) where p and q
are arbitrary maps such that if 00 is a loop in Z then pew) is contractible in Y, and obtain a Union Theoren similar to the above. § 3 , § 4 are the 1-dimensional versions of § 1 , § 2 respectively. In the last section we prove some results on colimits and the fundamental groupoid, useful for the next chapter.
Chapter 2 deals with the fundamental groupoid of a space BU ~ssociated to a cover U of X. This space has been considered by Segal who has proved that the projection p : BU ... X is a homotopy equivalence if U is numerable. Here we prove that p induces an equivalence of ftlntl81nental. eroupoids when the interiors of the ele::lents of U cover X I and we show how this is r(~llited to a theorem of 1,~acbeath-Sv;an relating ~(x) to G in the case the (discrete) group G acts on X •