Van Kampen theorems for diagrams of spaces

On five well-known commutator identities

“…https://doi.org/10.1017/S1446788700036934 [10] We prove Theorem 9 in Section 2. In [3] (and also in [5]) it is shown that for any group P with normal subgroup . /V such that H 2 (P) = H 3 (P) -0, there is an isomorphism…”

“…We attempt to obtain such an isomorphism but only succeed in proving injectivity on y n {F) for n = 1 , 2 , 3 . The proof uses a new description (in terms of the non-abelian tensor product defined in [3]) of the triple Pontryagin product Multiplicative Lie algebras are interesting algebraic objects in their own right, and so this paper also contains various results about them which are of purely intrinsic interest.…”

“…It is shown by topological methods in [3] (and algebraically in [5]) that if H 2 (P) = H 3 (P) = 0 then there is an isomorphism H 3 (P/N) s ker(iV A P -P). LEMMA …”

“…The proofs of Theorems 3 and 7 depend on the non-abelian tensor and exterior products introduced in [3] (see also [2]). For convenience we recall these constructions here.…”

The Seifert-Van Kampen Theorem

“…It is also clear that f is the only homomorphism that makes these diagrams commutative. Brown et al(1987) give a precise statement of the theorem. We assume that U and V are arcs/wise-connected open subsets of X such that X = U  V and U  V is non-empty any arcwise-connected.…”