Timeline for weak equivalence of simplicial sets

6 events

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Apr 29, 2011 at 16:14	comment	added	Allen Hatcher		A counterexample is given in Example 4.35 in the textbook that's mentioned in the question. It's a pretty simple construction: Start with $S^1\vee S^n$, $n >1$, and attach an $(n+1)$-cell by a map $S^n \to S^1 \vee S^n$ representing the element $2t-1$ in $\pi_n(S^1 \vee S^n) = {\Bbb Z}[t,t^{-1}]$. Then the inclusion of $S^1$ into the resulting space is an isomorphism on all homology groups and on $\pi_i$ for $i < n$ but not on $\pi_n$.
Apr 28, 2011 at 8:21	comment	added	Johannes Ebert		In my answer to this question: mathoverflow.net/questions/53399/…; I gave an example of a map which is a homology equivalence and an isomorphism on $\pi_i$ for $i < n$ (for a fixed $n$ that can be arbitrary large). Moreover, all homotopy groups of these spaces are abstractly isomorphic. You can see what goes wrong in the Leray-Serre spectral sequence. Talking about fibrant simplicial sets in this situation is a distraction.
Apr 28, 2011 at 4:04	comment	added	Somnath Basu		It seems that this question (and questions of this kind) can be answered generally using usual topological spaces. Using words like simplicial sets, morphisms and fibrant objects may be overkill!
Apr 27, 2011 at 19:54	answer	added	John Klein		timeline score: 7
Apr 27, 2011 at 19:50	history	edited	Enxin Wu		edited tags
Apr 27, 2011 at 19:16	history	asked	Enxin Wu	CC BY-SA 3.0