论局部乘积与POINCARE-ALEXANDER-LEFSCHETZ型对偶定理

<正> 在§1我们界说了局部乘积,它关联 Hausdorff 紧致空间 X 中闭子集X_0的同调以及 X_0在 X 中邻域的同调.在流形与有边流形上的 Poincaré-Alexander-Lefschetz 型对偶定理可以用这种局部乘积表示(§2).在§3,我们研讨了一类所谓摹流形状空间.局部的下调群与上调群的概念在 3,233—263页;8]中曾不明显地使

A NOTE ON LOCAL PRODUCTS AND DUALITY OF POINCARE-ALEXANDER-LEFSCHETZ TYPE

We define in § l the local products,which relates the homology of aclosed subset X_0 of a compact Hausdorff space X and that of the neigh-bourhoods of X_0 in X.Duality theorems of Alexander,Lefschetz and Poin-carétype in manifolds and manifolds with boundary are expressed in termsof these local products.In § 3,we study a class of manifold-like spaces.The use of local homology or cohomology groups was implicit in3,pp.233—263;8].Products given in the“local”way seem natural and useful.Let X be a compact Hausdorff space and X_0,E closed subsets of X·Let E_0=X_0∩E and let G be a coefficient group.Local homology groupsH_i(X_0|X,E_0|E;G)are defined as limits of direct system of relative homo-logy groups of the pairs(X,(X—W)U E)where W are neighbourhoodsof X-0 in X.Local cohomology groups H~i(X_0|X,E_0|E;G)are defined aslimits of inverse systems of relative cohomology groups of such pairs.Localcup and cap products are established in a natural way asH~i(X_0,G_1)(?)H~j(X_0|X,E_0|E;G_2)→H~(i+J)(X_0|X,E_0|E;G_0),H~i(X_0,G_1)⌒H~(i+j)(X_0|X,E_0|E;G_2)→H_j(X_0|X,E_0|E;G_0),H~i(X_0|X,E_0|E;G_1)⌒H_(i+j)(X_0|X,E_0|E;G_2)→H_j(X_0,G_0),where G_1 and G_2 are given coefficient groups paired to a coefficient gorupG_0.We summarize here some of the results.For instance,(i)if X is anorientable n-dimensional manifold with regular boundary E(n≥1),and if(?)(X)is an integral fundamental homology class of X,then local cap pro-ducts H~i(X_0|X,E_0|E;G)⌒(?)(X)etc.yield isomorphisms⌒(?)(X):H~i(X_0|X,E_0|E;G)≈H_(n-i)(X_0,G)etc.When X_0=X,this reduces to the ordinary duality of Poincaré-Lefschetztype.(ii)If X is of finite dimension and satisfies suitable local homology properties,then,with coefficients in a field(?),the following three condi-tions(a),(b),(c)are equivalent:(a)There is an element (?)(X)of H_n(X,(?))such that⌒(?)(X):H~i(X_0,(?))≈H_(n-i)(X_0|X,(?))for every closed subset X_0 of X and every integer i.(b)There is an element(?)(X)of H_n(X,(?))such that⌒(?)(X):H~i(X_0|X,(?))≈H_(n-i)(X_0,(?))for every closed subset X_0 of X and every integer i.(c)For every closed subset X_0 of X and every integer i,(?)is a dual pairing.(iii)If X is a non-empty connected compactum,which is lc~1 over(?),and if there is(?)(X)∈H_2(X,(?))such that(?)for every closed subset X_0 of X and for 0≤i≤2,then X is a closed sur-face.This paper was completed in the United states of America.Abstractappears also in a few lines in the notes“periodic transformations andfixed point theoremsⅠ,Ⅱ”,Science Record,Vol.1(1957),No.1.