The homology groups of spaces which the paper will consider,all assumedto have finite generators,and the homology group means singular homo-logy group with intger coefficients.Let Q be a set of positive primenumbers,then C_Q denotes the class of all the abelian groups,the order ofany element of which is finite,and all of its prime factors are containedin Q.Let C_f denote the class of abelian groups of finite generators,thenwe have the following theorem:Theorem 1.For any class of abelian group(for the definition,see[15]),one of the following results holds,and only one holds:(a)C_f(?)C(b)There exists a set Q of positive prime numbers such that(?)Let C be any class of abelian group.An arcwised connected topologicalspace is said to be n-C-connected,if ∏_1(X)=0 and ∏_i(X)∈C when2≤i≤n.Let Φ_n~X denote the natural homorphism from ∏_n(X)to H_n(X)for any positive number m.Let S(m)denote the set of positive primerumbers such that 2p-3≤m,then we have the following relations betweenhomotopy groups and homology groups.Theorem 2.(The main theorem of the paper.)Let X be a(n-1)-C_Q-connected space.Then:(a)when n≤m≤2n-2, Φ_m~X is a C_((QUS)(m-n))-bi-unique andC_((QUS)(m-n-1))-onto homorphisms.(For the definition of C-bi-unique homor-phism and C-onto homorphism,see[15].)(b)When m=2n-1,Φ_m~X is a C_((QUS)(m-n-1))homorphism.The proof oftheorem 2 is thus:At first,proving this theorem for the A_n~1 polyhedra∑~n for which(?),for any(n-1)connected space X,usingHurwicz theorem and mapping cylinder,we may assume that there existsone A_n~1 polyhedra ∑~n such that H_(n+1)(∑~n)=0 and the homorphisms ofH_n(∑~n),∏_n(∑~n)into H_n(X),∏_n(X)induced by the injection of ∑~ninto X,are isomorphisms onto.Let X be the space of all paths which start at a fixed point P in X,and ∑~n be the space of paths which startat P and end at points of ∑~n,then using induction and relations betweenH_m(x),∏_m(x),H_m(X,∑~n),∏_m(X,∑~n),H_m(X,∑~n),∏_m(X,∑~n),H_m(∑~n)and ∏_m(∑~n),we may prove this theorem for any(n-1)--connected space,X.For any n-1--C_Q--connected space X,by the me-thod of Cartan-Serre-Whitehead,we may construct a(n-1)--connectedspace X′and a mapping of X′into X,such that the homorphisms inducedby F of H_m(X′),∏_m(X′)into H_m(X),∏_m(X)all are C_Q isomorphisms,thenusing the relations between H_m(X′),∏_m(X′),H_m(X)and ∏_m(X),we mayprove theorem 2,for any(n-1)--C_Q--connected space.Let τ_n~(n+1)denote the non-zero element of ∏_(n+1)(S~n)and 0 denote the com-position defined in[16],then we have the following theorem:Theorem 3.Let X be a(n-1)connected space n≥3,then,∏_n(X)~(-1)0τ_n~(n+1)is the Kernel of Φ_(n+1)~X.Let ∏_m(G,n)denote the m-th homotopy group of space of homologytype(G,n)(for the definition of space of homology type(G,n)see[13]),then we have the following theorem:Theorem 4.If a simply connected space X satisfied the following con-ditions:H_m(X)=0 when 0
