<正> 并于复合形或更一般的空间在欧氏空间中的实现问题,曾经有过下面几个重要的结果:1°Van Kampen 在1932时证明有不能在2n维欧氏空间中实现的 n 维复合形 K 存在。Van Kampen 的证明倚赖于由及的约化二重对称积 K 作出的一个不变量.作者在[2]中指出 Van Kampen 不变量只是一组不变量(?),(I_m=整数加法群 I 或法2整数群 I_2,视 m 为偶或奇而定)其中极端的一个,即Φ~(2n),而Φ~m=0为

For the realization problem of complexes or more general spaces in eu-clidean spaces we may cite the following results:1°In 1932 Van Kampen~[1]proved the existence of n-dimensional comp-lexes K not realizable in euclidean space of dimension 2n.His proof depends on certain invariant deduced from the reduced 2-foldsymmetric product K of K.The author~[2] has pointed out that Van Kampen'sinvariant is merely the extreme one of a system of invariants Φ~m∈H~m(K,I(m)).(I(m)=I or I_2,depending on m being even or odd),namely Φ~(2n),andΦ~m=0 is a necessary condition for K to be realizable in R~m.We provedalso in[2] the topological invariance of Φ~m,i.e.,the independence of Φ~m fromthe chosen subdivision K of the space of K,while Van Kampen,in the ex-treme case m=2n considered by him,has not even proved its combina-torial invariance.2°In the case of differentiable manifold,Whitney(cf.e.g.[3])hasintroduced a system of invariants which I have called the dual Whitneyclasses(?),and proved that(?)=O,k≥m-n(1)are necessary conditions for an n-dimensional closed differentiable manifoldto be realizable in R~m.3°For compact Hausdorff spaces X,Thom has proved([4]Th.Ⅲ.25)the following theorem:Let Q~i be certain operations deduced from Steenrodsquares,thenQ~iH~r(x,l_2)=0,2i+r≥m(2)are necessary conditions for X to be realizable in R~m The operations Q~iwas previously introduced by the author by applying Smith's theory of pe-riodic transformations and will henceforth be denoted by Sm~i,cf.[5]and [6].Besides,Flores~([7])get also same results as Van Kampen concerning theexistence of n-dimensional complexes K not realizable in R~(2n).However,whathe used to prove his results is get by imbedding first K in R~(2n+1),and isactually not an invariant but depends in general on the way of realization of K in R~(2n+1).This will be explained further in the sequel.In the various theories listed above,not only the methods used are quitedifferent from each other,but also the objects and the realization conceptinvolved are not the same.For example,the theory of Van Kampen studiesabout the semi-linear realization of finite complexes,the theory of Whitneyis applicable only for the differentiable realization of differentiable mani-folds,and the theory of Thom is concerned about topological realization ofmore general spaces.The present-paper gives a general theory including the various theoriesabove cited as its particular cases.This theory may be formulated in termsof the following easily proved fundamental theorem:For any Hausdorff space X let X be the space of all ordered pairs(x_1,x_2),where x_1,x_2 ∈X and x_1≠x_2.Let t:(?)≡(?)be the transformationt(x_1,x_2))=(x_2,x_1),X the modular space (?)/t.With respect to the pair((?),t),we may define according to P.A.Smith~([8])a system of cohomologyclassesΦ~m(X)∈H~m(X,I(m)),where H represents singular homology system.Then Φ~m(X)=0 is a nece-ssary condition for X to be topologically realizable in R~m.If X be a finite polyhedron,then according to results of[2],the classesΦ~m defined above are the same as the classes Φ~m defined in that paper ina quite different manner.It follows that Φ~m=0 is not only a necessarycondition for X to possess a subdivision K such that K is semilinearly rea-lizable in R~m,as shown in[2],but also one for X itself to be topologicallyrealizable in R~m.If X is a closed differentiable manifold of dimension n,then we provethat from ρ_2Φ~m(X)=0 we get(1),but not vice versa.Our fundamentaltheorem has therefore Whitney's theorem as its consequence.It followsalso that Whitney's condition(1)is not only necessary for the differentiablerealization,but also necessary for topological realization of X in R~m.We prove also that if X be a finite polyhedron,then from ρ_2Φ~m(X)=0we get(2).Hence our fundamental theorem has also,at least in the caseof finite polyhedron,Thom's theorem as its consequence.In the case of differentiable manifold,Whitney~([9])has introduced theconcept of“immersion”in a euclidean space.He proved that any differen-tiable manifold of dimension n may be immersed in R~(