A Van Kampen theorem for π2

AbstractIf the path connected topological space X has a countable open cover U with path connected elements, then π2(X,∗) is computed as a colimit determined by the second homotopy groups of the intersection of elements of U and the indices of the fundamental group injections of these intersections into the fundamental group of X. Aside from assuming that the inclusions induce such monomorphisms, certain other inclusions are also required to induce monomorphisms of fundamental groups and restrictions are placed on the arrangement of the elements of U