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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The singular cohomology of the inverse limit of a Postnikov tower is representable


Authors: Jerzy Dydak and Ross Geoghegan
Journal: Proc. Amer. Math. Soc. 98 (1986), 649-654
MSC: Primary 55S45; Secondary 54E60, 55P55
MathSciNet review: 861769
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Abstract: Let $ {X_1} \leftarrow {X_2} \leftarrow \cdots $ be an inverse sequence of spaces and maps satisfying (i) each $ {X_n}$ has the homotopy type of a CW complex, (ii) each $ {f_n}$ is a Hurewicz fibration, and (iii) the connectivity of the fiber of $ {f_n}$ goes to $ \infty $ with $ n$. Let $ \hat X$ be the inverse limit of the sequence. It is shown that the natural homomorphism $ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} ^k}(\hat X,G) \to {H^k}(\hat X,G)$ (from Čech cohomology to singular cohomology, with ordinary coefficient module $ G$) is an isomorphism for all $ k$. It follows that $ {\lim _{ \to n}}[{X_n},K(G,k)] \cong [\hat X,K(G,k)]$ for any Eilenberg- Mac Lane space $ K(G,k)$. It is also shown that, except in trivial cases, $ X$ does not have the homotopy type of a CW complex.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0861769-8
PII: S 0002-9939(1986)0861769-8
Keywords: Čech cohomology, Postnikov system, Homotopy type of CW complex
Article copyright: © Copyright 1986 American Mathematical Society