Duality in manifolds
The main purpose of this paper is to establish a very general Alexander duality theorem for pairs (X,A) of subsets of compact orientable n- dimensional manifolds M. Homology and cohomology groups are taken with coefficients in a spectrum \({\mathcal E}=\{E_ n\), \(\sigma_ n: \Sigma E_ n\to E_{n+1}\}\). For compact metric X homology groups are defined by \({\mathcal E}_ n(X)=co\lim_ k[S^{n+k},E_ k\wedge X]\). For arbitrary k-spaces homology is defined by taking the direct limit over compact subsets. Cohomology groups of a space X are defined by \({\mathcal E}^ n(X)=\lim_ k[\Sigma^ kX,E_{n+k}]\). In this definitions [X,Y] does not denote the set of homotopy classes of maps \(X\to Y\) as in the classical paper of \textit{G. W. Whitehead} [Trans. Am. Math. Soc. 102, 227- 283 (1962; Zbl 0124.383)]; instead, it denotes the set of strong shape morphisms \(X\to Y\). In order to define smash products of shape morphisms, the author had to enlarge his original strong shape category \({\mathcal K}_ h\) [Pac. J. Math. 64, 25-65 (1976; Zbl 0346.55014)] by adding to it new objects. In the original category objects were topological spaces X identified with certain 2-categories \({\mathcal P}_ X\) of maps \(X\to P\) into ANR's P. In the present paper the additional objects are all finite ''smash products'' \(X_ 1{\bar \wedge}...{\bar \wedge}X_ n\) defined as certain 2-categories of maps \(X_ 1\wedge...\wedge X_ n\to P\) of the usual smash product factorized through smash products \(P_ 1\wedge...\wedge P_ n\) of ANR's. For \(X\subseteq S^ n\) the obtained duality theorem gives a natural isomorphism \({\mathcal E}_ p(X,x_ 0)\approx {\mathcal E}^ q(S^ n\setminus \{x_ 0\},S^ n\setminus X)\approx \tilde {\mathcal E}^{q-1}(S^ n\setminus X)\), \(p+q=n\). In the non-compact case it is assumed that \({\mathcal E}\) is an ANR-spectrum. The theorem generalizes to \({\mathcal F}\)- orientable compact n-manifolds M, where \({\mathcal F}\) is a ring spectrum of ANR's and \({\mathcal E}\) is an \({\mathcal F}\)-module spectrum. In the relative case it assumes the form \({\mathcal E}_ p(X,A)\approx {\mathcal E}^ q(M\setminus A,M\setminus X)\), \(p+q=n\). The author proceeds to show that his theorem contains as special cases other known duality theorems. In particular, the Steenrod-Sitnikov duality [see \textit{P. S. Aleksandrov}, Tr. Mat. Inst. Steklova 54, 1-36 (1959; Zbl 0092.154)] and the duality theorem of \textit{D. S. Kahn}, \textit{J. Kaminker} and \textit{C. Schochet} [Mich. Math. J. 24, 203-224 (1977; Zbl 0384.55001)]. To prove the first assertion, the author shows that for finitely generated Abelian groups G the homology \(H_ p(X;G)\), introduced in his paper [loc. cit.], agrees with the homology \({\mathcal E}_ p(X)\) with coefficients in the Eilenberg- MacLane spectrum \({\mathcal E}=\{K(G,n)\}\). However, for compacta which are not pointed 1-movable, \(H_ p(X;G)\) needs not to agree with the corresponding Steendrod-Sitnikov homology group (as an example of \textit{A. Koyama} shows [Proc. Japan Acad., Ser. A 60, 319-322 (1984; Zbl 0559.55006)]). How to derive in such cases the Steenrod-Sitnikov duality from the author's duality remained unclear to the reviewer.
Alexander duality theorem for compact orientable n-dimensional manifolds
cohomology groups with coefficients in a spectrum
scientific article; zbMATH DE number 3899736