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Isomorphic cohomology yields isomorphic homology. (English) Zbl 0886.55005

Given a map \(f:X\rightarrow Y\) of locally compact Hausdorff spaces, W. S. Massey [Algebraic topology: an introduction (1967; Zbl 0153.24901)] has shown that if \(f\) induces an isomorphism \(H^n(X;{\mathbb{Z}})\cong H^n(Y;{\mathbb{Z}})\) for all \(n\geq 0\), then \(f\) induces an isomorphism in homology with coefficients in any abelian group. The authors generalize this result to show that for any non-complete principal ideal domain \(R\), a map \(f:X\rightarrow Y\) of chain complexes of \(R\)-modules that yields an isomorphism in cohomology with coefficients in \(R\) also produces an isomorphism in homology with coefficients in any \(R\)-module. This result is obtained by proving that if \(\operatorname{Hom}_R(M,R)=0\) and \(\text{Ext}_R(M,R)=0\), then \(M=0\) for any \(R\)-module \(M\). The authors use the main result to obtain a version of the Dual Whitehead Theorem due to H. J. Baues [Obstruction theory on homotopy classification of maps, Lect. Notes Math. 628 (1977; Zbl 0361.55017)]. In particular, they establish that a map \(f:X\rightarrow Y\) of \(R\)-Postnikov spaces of order \(k\geq 1\) that induces an isomorphism in cohomology must be a weak homotopy equivalence.

MSC:

55N10 Singular homology and cohomology theory
55U20 Universal coefficient theorems, Bockstein operator
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
13F10 Principal ideal rings
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References:

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