複合形在歐氏空間中的實現問题Ⅰ

<正> 在拓撲發展之初很早就知道一個抽象的n維單純複合形(有限或無限)必可在2n+1維歐氏空間及R~(2n+1)中得到實現,它的證明也很簡單(例如見[1]§2或[2]第Ⅲ章§2).從這一定理知道2n+1維的歐氏空間實際上已包括了所有想像得到的n維複合形,可是是否有不能在R~m中實現但能在R~(m+1)中實現的

ON THE REALIZATION OF COMPLEXES IN EUCLIDEAN SPACES I

It was early known that any n-dimensional abstract complex may be realized in an (2n + 1) -dimensional euclidean space R~(2n+1). From this theorem, whose proof is quite simple, it follows that the (2n + 1)-dimensional euclidean space contains in reality all imaginable n-dimensional complexes. However, the complete recognization of all n-dimensional complexes in an euclidean space of given dimension m where m<2n+1 is a problem much more difficult which cannot, it seems, be solved completely in the near future. Among the miscellaneous results so far obtained along this line the most remarkable one is no doubt that of Van Kampen and Flores, who first proved the existence of n-dimensional complexes which, even under further subdivisions, cannot be realized in an R~(2n).The invariant by means of which Van Kampen was able to conclude the non-realizability of a (finite simplicial) n-dimensional complex in an R~(2n) may be described as follows. Denote the k-dimensional simplexes of the given n-dimensional complex K by S_i~k. Any two simplexes of K with no vertices in common will be said to be disjoint. Let A be the set of all unordered index pairs (i,j), corresponding to pairs of disjoint n-dimensional simplexes S_i~n and S_i~n. Construct a vector space on the ring of integers with dimension equal to the number of elements in A. Any vector of may then be represented by a system of integers (a_(ij)) where (i, i)∈ A. To each pair of disjoint simplexes S_a~(n-1) and S_l~n in K a certain vector V_(la)= (a_(ij)) of may be determined in the following manner. If both i, j≠lor one of them, say j=l, but S_a~(n-1) is not a face of S_i~n, then we put a_(ij)= 0. Otherwise we put a_(il) = ±1 (with sign conveniently chosen). Two vectors P, P of will then be said to be equivalent if P-P is a certain linear combination with integral coefficients of vectors of form V_(la) above defined. The vectors of are thus distributed in such equivalence classes.Take now an arbitrary simplicial subdivision K_1 of K and try to realize K_1 in R~(2n) as much as possible. We will obtain then some "almost true" realization such that parts ′S_i~k and ′S-i~l, corresponding to disjoint S_i~k and S_i~l of K will be disjoint in R~(2n) when k + l < 2n, while they intersect only in isolated points when k=l=n. With respect to an orientation arbitrarily chosen of R~(2n), 'S_i~n and 'S_i~n determine then a definite intersection number ±a_(ij) (with sign conveniently chosen). These numbers determine in turn a vector P = (a_(ij)) of Van Kampen's work shows that, whatever be the subdivision K_1 of K and the "almost true" realization of K_1 in R~(2n), the corresponding vectors P belong always to one and the same equivalence class in It follows that this equivalence class is an invariant of the complex K. It is evident that the belongness of the zero vector to this invariant equivalence class is a necessary condition for the existence of "true" realization of K in R~(2n). It is this invariant which has enabled Van Kampen to assure the existence of n-dimensional complexes non-realizable in R~(2n). On the other side, Van Kampen failed to ascertain whether the above necessary condition is also sufficient and the problem of characterizing n-dimensional complexes in R~(2n) remains unsettled up to the present. Moreover the method of Van Kampen-Flores cannot be seen to be readily generalizable to the realizability in R~m, m being arbitrary. We remark also that whether Van Kampen's invariant is a "topological" invariant of the space of K, or even whether it is a combinatorial invariant of K_1 cannot be decided from his works.At the time of Van Kampen and Flores the cohomology theory has not yet been created. To get a deeper insight of their results we will reformulate them in the modern terminology of cohomology. Their statements will then become clear and natural as follows. From the given simplicial complex (of any dimension) let us construct a subcomplex K of K×K, consisting of all cells σ×τ such that σ, τ are disjoint in K. Identify each pair σ×τ and τ×σ of K to t