Complement of lines and wedges of spheres - MathOverflow 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 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 Answer by jc for Complement of lines and wedges of spheres 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>