论附属于某种连续映像的不等式及其在纤维空间中的应用

<正> §1. 设 X,Y 为拓扑空间,又设 f:X→Y 为连续映像.J.H.C.Whitehead 证明 X,Y 为 CW 丛而 f 能导出基本群及上同调群间的同模对应时,f 为同伦对等映像.映像 f 是否存在,不仅与 X,Y 的基本群及上同调群的构造有关,而与 X,Y 内在的几何结构有密切的关系.连续照像 f 导出 X,Y 之间上同调群的准同模对应 f,那末 f 能与某些准同模对应相交换,由此 J.H.C.Whitehead 指出正则准同模的观念.由4]可知正则同模论供应我们许多同伦不变量,它们是直接可以计算的东西,并且对于 X,Y 间连续映像的分类问题,应该有密切的关系.

ON INTRINSIC INEQUALITIES ASSOCIATED WITH CERTAIN CONTINUOUS MAPPINGS AND THEIR APPLICATION TO FIBER SPACES

Between two polyhedra X and Y if there exists a map,f:X→Y,whichinduces isomorphisms into of cohomology groups of Y with respect to any givencoefficient group into the corresponding cohomology groups of X,then there mustbe some“internal”properties between X and Y.Our purpose is to express theseproperties by means of algebraic relations.Of course we may also consider theproblem if f induces such homomorphisms f:H~(Y, G)→H~n(X,G)which arealways onto.Due to the development of theory of proper isomorphisms we havehad a number of easily computable homotopy invariants associated with a polyhedron.The simplest among those invariants are block invariants.In this paper we givecomplete exposition of inequalities between block invariants of X and Y if thereis a map f:X→Y such that f:H~n(Y,G)→H~n(X,G)is always isomorphism intoor homomorphism onto for all n and G.The results are applicable to fiber theorybecause the fiber map p:E→B induces homomorphisms p:H~n(B,G)→H~n(E,G)which are always isomorphisms into if there is a section.An outline of this paperhas appeared in Science Record,Academia Sinica,1958.