Is there anything good in the class of objects with trivial higher homotopy and homology groups? Can it be described in some terms?
For example: such $X$ that $\pi_{\gg 0}(X) = 0$ and $H_{\gg 0}(X,\mathbb{Z})=0$
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3$\begingroup$ What do you mean by "anything good"? There are some useful examples, such as $S^1$ and any closed connected surface that isn't $S^2$ or $\mathbb{RP}^2$. $\endgroup$– Arun DebrayCommented Apr 17, 2017 at 12:53
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2$\begingroup$ And there are wild things like Cantor sets, too. Also, what kind of answer are you expecting? Please be a bit more precise. $\endgroup$– Sebastian GoetteCommented Apr 17, 2017 at 13:01
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3$\begingroup$ What does $\pi_{\gg 0}$ denote? Does it refer to $\pi_n$ for $n \geq 1$, $n \geq 2$, or $n \geq k$ for some choice of $k$? $\endgroup$– Michael AlbaneseCommented Apr 17, 2017 at 14:00
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$\begingroup$ Under these conditions, the Postnikov tower will be finite. So the homology groups will be determined by the homotopy groups together with certain cocycles in the homology of the terms of the tower with coefficients in the homotopy groups. ncatlab.org/nlab/show/… It seems likely to be quite intricate to determine in general when the homology will be finite. $\endgroup$– Ian AgolCommented Apr 18, 2017 at 17:33
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1 Answer
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One thing you should bear in mind is the Kan-Thurston theorem: if $X$ is connected then there is a group $G$ and a map $f\colon BG\to X$ such that the induced map $H_*(BG)\to H_*(X)$ is an isomorphism. Thus, if $H_i(X)=0$ for $i\gg 0$ then the space $BG$ will have $H_i(BG)=0$ for $i\gg 0$ and also $\pi_i(BG)=0$ for $i>1$. Of course, the groups $G$ that arise in this context are typically quite strange and unfamiliar.
answered Apr 17, 2017 at 13:19
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$\begingroup$ Sorry but, what is $BG$ ? $\endgroup$ Commented Apr 17, 2017 at 13:41
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4$\begingroup$ $BG$ is the classifying space of the group $G$. It is a space characterized up to homotopy by having the homotopy type of a CW-complex, being connected, having $\pi_1(BG,x) \cong G$ for any choice of basepoint, and all higher homotopy groups trivial. $\endgroup$ Commented Apr 17, 2017 at 14:44
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2$\begingroup$ Ian Leary gave an improved version where $BG$ is a non-positively curved polyhedral complex. (Maybe even a cube complex?) So the groups $G$ that arise needn't be that "strange and unfamiliar" -- they have solvable word problem, for instance. $\endgroup$– HJRWCommented Apr 18, 2017 at 6:00