同伦群和同调群的关系及其应用

<正> 最近几年来,关于同伦群的研究,特别是球面同化群的计算有很大的进步,但是关于更一般的空间的同伦群的计算,就作者所知道的文献来说,似乎没有相应的进展,在这一方面的工作,最早也是最主要的是 Hurewicz 定理,Serre 在15]中利用所谓 C同构的概念,把 Hurewicz 定理推广到所谓(n-1)维 C 连通空间中,但是也和 Hure-wicz 定理一样,只能讨论 n 维同伦群和 n 维同调群的关系.张素诚,Hilton 及 Barrat

THE RELATIONS BETWEEN HOMOPTOPY GROUPS AND HOMOLOGY GROUPS,AND SOME OF ITS APPLICATIONS

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,see15]),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,see15].)(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 in16],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)see13]),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