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Disclaimer: What follows is probably a bit off-topic for this site, but no more than the original questions, numbered one and two. In fact I suspect that this answer attempts to address just what the …

Maybe I should put the question this way: is or is not PL topology an integral part of the education of every geometric topologist today? According to a recent poll by the Central Planning Commit …

Geometric topology is not really a unique field in the way that algebraic topology and general topology are. Its various subareas may share something of a common feel (and indeed an arxiv category), b …

What you describe is similar to something well-known that does satisfy your condition (ie, has the same homotopy as $X$ in dimensions less than $n-k$). I'm speaking of the dual complex to the dual $(n …

How about $X=\Bbb RP^2\times L^2_3$, where $L^2_3$ is the cone of the $3$-fold cover $S^1\to S^1$. Here $H^4(X;\Bbb Z)\simeq\Bbb Z/2\otimes\Bbb Z/3=0$, but $H^4(X;\mathcal L)\simeq\Bbb Z\otimes\Bbb Z/ …

A senseful meaning of ‘approximation of manifolds’? A simply-looking meaning is the one of Gromov-Hausdorff, as mentioned by Ryan Budney. It has many cool applications, but none that I'm aware of ar …