## 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

### Citations:

Zbl 0153.24901; Zbl 0457.55001; Zbl 0361.55017
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### References:

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