In what follows all complexes will be understobd to be euclidean finitesimplicial complexes.Let K be a complex with space K and R~m a euclidean space of dimensionm.A topological mapping T:(?)→R~m will be called a linear imbedding of K inR~m if T is linear on each σ∈K.If T:(?)→R~m is a linear imbedding of a certainsimplicial subdivision K′of K in R~m,then T will be called a semi-linear im-bedding of K in R~m through the subdivision K′.A continuous mapping T:K→R~m will be called a linear pseudo-imbedding of K in R~m if the following conditionsare satisfied:1°.T is linear on each σ∈K,2°.T(σ)is a non-degenerate simplex ofsame dimension as σ∈K if dim.σ≤m,and 3°.for any pair a,σ∈K withdim σ+dim τ≤m,T(σ)and T(τ)are in general position.If T:(?)→R~m is alinear pseudo-imbedding of a certain simplicial subdivision K′of K in R~m thenT will be called a semi-linear pseudo-imbedding of K in R~m through the sub-division K′.For any K let(?)be the subcomplex of the product complex K×K formedof all cells σ×τ for which σ∈ K are disjoint,i.e.,having no vertices incommon.Let us identify each pair σ×τ and τ×σ of(?)into a single cell σ*τ(or σ*τ),then by orienting σ*τ as σ×τ,we have for the,complex K formedof all cells σ*τ,σ,τ∈K disjoint:(?) (1)(?) (2)(?) (3)The complex(?)is naturally a two-sheeted covering complex of K and wemay thus define for the pair((?),K),according to the general theory of P.A.Smith about periodic transformations,a system of classes Φ~m(K)∈H~m(K, I_((m))),m≥0,where I_((m))is the group Of integers I for m even and the group I_2 ofintegers mod 2 for m odd.In the same way,for any Hausdorff space X let(?)be the subspace of the product space X×X consisting of all points(x,y)forwhich x,y∈X and x≠y.By identifying each pair(x,y)and(y,x)in(?)weget a space X of which(?)is naturally a two-sheeted covering space.Withrespect to the pair((?),X)we may thus also define,by the theory of Smith,a system of classes Φ~m(X)∈ H~m(X,I_((m))),m≥0,which are,by definition,topological invariants of X(H denotes singular homology).As remarked in[2],if X=(?),then H~m(K,G)are canonically isomorphic to H~m(X,G)andunder these isomorphisms,Φ~m(K)correspond to Φ~m(X)so that we may writeΦ~m(K)= Φ~m(X)legitimately.Now let T:(?)→R~m be any semi-linear pseudo-imbedding of K in R~m whichalways exists.For any σ∈ K,T_σ is then a chain of R~m.Take a fixed orientationof R~m and let φ denote the index of intersection in this oriented R~m and ρ(m)either the identity or the reduction mod 2 according as m be even or odd.Thenowing to (2),(?)is unambiguously well defined for any σ*τ∈K of dimension m and gives riseto a cochain ψ_Τ of dimension m in K with coefficient group I_((m)).As provedin[1],ψ_Τ is always a cocycle whose class is independent of the semi-linearpseudo-imbedding T as well as the orientation of R~m chosen and coincides inreality to the class Φ~m(K)=Φ~m(?).It has also been proved that Φ~m(K)=0(or Φ~m(X)=0 for any space X)is a necessary condition for K(or X)to besemi-linearly(or topologically)imbeddable in R~m The purpose of the presentpaper is to prove the converse in the extreme case m=2n where n==dim K>2.In fact,we have the following. Theorem.For a finite simplicial complex K of dimension n>2 to be semi-linearly imbeddable in R~(2n)it is necessary and sufficient that Φ~(2n)(K)=0.We have also the following.Corollary.A finite simplicial complex K of dimension n>2 is semi-linearlyimbeddable in R~(2n)in each of the following four cases:1°.H_n(K,mod 2)=0.2°.H_n(K)=0.3°.K is an irreducibly closed complexor 4°.each(n-1)-dimensional simplex of K is the face of at most twon-dimensional simplexes of K.The proof of the sufficiency part in our theorem depends on the followingthree constructions.A.Tube construction.Let C be a differentiable simple arc in R~m with end points a_0,a_1.Let L_i~n(i=0,1)be two n-dimensional linear subspaces of R~m orthogonal to C at a_i,and S_i~n-1 the(n-1)-dimensional sphere of center a_i and radius ε>0 in L_i~n.Suppose that S_i~(n-1)have been oriented.
