most recent 30 from http://mathoverflow.net 2013-05-24T15:39:02Z http://mathoverflow.net/feeds/question/41522 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41522/complement-of-lines-and-wedges-of-spheres A. Pascal 2010-10-08T16:57:30Z 2010-10-08T18:01:18Z

<p>Let $L=L_1 \cup ... \cup L_n$ be the union of $n$ distinct lines through the origin in $\mathbb{R}^{3}$. I'd like a convincing argument that $\mathbb{R}^{3} \setminus L$ is homotopy equivalent to a wedge of $n$ circles (if that is true). In fact, I especially care about the case $n=2$.</p> <p>I know this sounds like a homework problem, but I have other purposes in mind (this space naturally showed up as the fibre in a certain fibration) and I don't find typical text-book explanations of such problems very convincing, so I would appreciate a clear answer.</p>

http://mathoverflow.net/questions/41522/complement-of-lines-and-wedges-of-spheres/41529#41529 jc 2010-10-08T17:47:55Z 2010-10-08T18:01:18Z

<p>First, deformation retract $\mathbb{R}^3$ minus $L$ to $S^2$ minus $2n$ points (you can do this since you've removed the origin). Stereographically project from one of the punctures, and you've got $\mathbb{R}^2$ minus $2n-1$ points. Choose a point away from the punctures and draw disjoint based loops around each of the remaining holes. Now deformation retract to those loops and you've got a wedge of $2n-1$ circles. </p>