The second van-Kampen theorem for topological spaces

AbstractLet X be a pathwise connected topological space and let X1 and X2 be two closed pathwise connected homeomorphic subspaces. An elementary proof is given that, provided X1 and X2 have disjoint open neighbourhoods and one of the spaces X1 and X2 is a strong deformation retract of its neighbourhood, the relation between the fundamental group of X and the space obtained by identifying X1 and X2 is the same as that discovered by van Kampen for less general conditions as a generalisation of an earlier theorem of Seifert. The relation also holds for nonclosed X1 and X2, if X is covered by the interior points of the complement of X1 ∪ X2 and the neighbourhoods of these spaces. Two counter-examples for this relation showing that local conditions are required are also presented in this paper. Also, this paper contains a discussion of additional properties for a neighbourhood deformation retract X1 that ensure that “collapsing a handle defined as trivial I-bundle” will induce a (weak) homotopy equivalence and it discusses relevant counter-examples