X Serre's result for compact  has been extended to arbitrary finite H - spaces in the combined work of Browder  and Kumpel simply connected finite H -space with the type of X.
RiA~ Ap This induces the commutative diagram ... (G;FiA/Fi+IA) ~TX -+H (G;Fi+IA) ie ~H (G;FiA) localize. el, e4 1e -+H(G;FiA/Fi+IA) -~H I(G;Fi+IA) n ~... (G ;r" A ) ~ n+l P P P n P P n P P n p p p n-~ p p " Here we know that n localize and we may assume inductively that It thus follows that e3 localizes. e2' e5 This completes the inductive step and establishes the lemma. We are now ready to prove that (ii) = (iii) in Theorem 4B. 2) Let 37 Since f induces localization in homotopy, it induces localization in homology by Theorem 2B.