A Theorem on Injectivity of the Cup Product

John C. Wood
1973 Proceedings of the American Mathematical Society  
We prove that if a space A'has abelian or sufficiently abelian fundamental group, then the cup product H1(X)f\H1(X)-* H2(X) is injective, giving an inequality between the associated Betti numbers. This generalises to a theorem of injectivity of the £-fold cup product on H"(X), given that the kth order Whitehead product on trn(X) is trivial or torsion. Let X be a topological space, and let Hn(X) denote its «th singular integral homology group. Let s/ denote the class of groups 7r such that
more » ... 7r such that p:TT-+TT¡[ir,Tr] splits rationally, i.e. there exists a homomorphism q:TTl[rr,-rr]-^-Tr such that pq®\ :-n\[ir, 7r]®ß-*ir/[7r, n]®Q is an isomor-
doi:10.2307/2038752 fatcat:la7insywizf3heaane6uxsqw4i