Duality in manifolds
type: Item
instance of: scholarly article
title: Duality in manifolds
zbMATH Open document ID: 0564.55001
DOI: 10.1007/BF01773386
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1984-01-01T00:00:00Z
review text: 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 \(…
Mathematics Subject Classification ID: 55M05
Mathematics Subject Classification ID: 55N20
Mathematics Subject Classification ID: 55N07
Mathematics Subject Classification ID: 55P55
zbMATH DE Number: 3899736
zbMATH Keywords: Alexander duality theorem for compact orientable n-dimensional manifolds
zbMATH Keywords: cohomology groups with coefficients in a spectrum
zbMATH Keywords: strong shape morphisms
zbMATH Keywords: smash products of shape morphisms
zbMATH Keywords: 2-categories of maps
zbMATH Keywords: ring spectrum of ANR's
zbMATH Keywords: Steenrod-Sitnikov duality
author: Friedrich Wilhelm Bauer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3899736