I am currently reading Dyer's Cohomology Theories and he in the very beginning makes the assumption to work in the category of finite CW complexes. This kills of many interesting objects ($K(G,n)$, $\mathbb{CP}^{\infty}$,$\Omega S^n$ etc...). A lot of interesting object also survive. I am wondering if there is any good reason to work in this category. Are there any theorems that only work for finite CW complexes and cannot be extended?
Differences between finite and infinite CW complex categories
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algebraic-topology
soft-question
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0At an introductory level, cohomology is often touted as being superior to homology because it carries the cup product. But in fact the real thing one ends up caring about is the fact that $E^*X$ is a module over the algebra $E^*E$ of cohomology operations, which (in many good cases) is dual to the coalgebra of "homology cooperations" $E_*E$ (over which $E_*X$ will be a comodule). This allows one to flip everything and work with homology, which totally sidesteps these finiteness issues since homology is better-behaved w/r/t infinite colimits. E.g. the Adams SS can be phrased either way. – 2011-09-27
1 Answers
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A CW complex if finite if and only if it is compact. There are innumerable things that hold for compact spaces but which are false (or at least more subtle) for noncompact spaces. For example, noncompact manifolds are far more complicated than compact manifolds.
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0$F$ound a good r$e$ason: the axioms for cohomology are simpler with finite CW complexes. – 2011-09-26