Question about the fundamental group and homotopy equivalence - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T19:17:09Z http://mathoverflow.net/feeds/question/31414 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/31414/question-about-the-fundamental-group-and-homotopy-equivalence Question about the fundamental group and homotopy equivalence Italo 2010-07-11T14:22:33Z 2010-07-11T16:16:28Z <p>Let T be a two-dimensional torus and Y be the one point compactification of a two dimensional sphere ($S^2$) minus three points. I have to prove:</p> <p>1)they have the same fundamental group</p> <p>2)they are homotopically equivalent</p> <p>This is what i thought: i can see Y this way, let A,B,C three distinct points on the sphere and P a point not on the sphere, Y is homotopically equivalent to $S^2\cup\overline{AP}\cup\overline{BP}\cup\overline{CP}$. My hope for showing that they have the same fundamnetal group was to use van kampen theorem with open sets one $U$ homeomorphic to an open disk and the other one $V$ homotopically equivalent to a bouquet of 2 circles and $U\cap V$ homotopically equivalent to a circle. Now i would like to show that the generator of $\pi_1(U\cap V)$ is sent by inclusion map to the commutator of the generators of $\pi_1(V)$ but i can't see how.</p> <p>For the homotopy equivalence i can't see the homotopy between these two spaces.</p> <p>Please can anyone help?</p> <p>Thank you in advance.</p> http://mathoverflow.net/questions/31414/question-about-the-fundamental-group-and-homotopy-equivalence/31415#31415 Answer by Gregory Arone for Question about the fundamental group and homotopy equivalence Gregory Arone 2010-07-11T14:36:38Z 2010-07-11T14:36:38Z <p>I think Y is homotopy equivalent to $S^2\vee S^1\vee S^1$. Proof: $Y$ is homotopy equivalent to the homotopy cofiber or the map from $3$ point to $S^2$. This map is null-homotopic, so $Y$ is equivalent to the wedge sum of $S^2$ with the suspension of three points. Suspension of three points is equivalent to $S^1\vee S^1$. </p> <p>It is easy enough to construct the homotopy equivalence directly.</p> <p>In conclusion: both (1) and (2) are false.</p> http://mathoverflow.net/questions/31414/question-about-the-fundamental-group-and-homotopy-equivalence/31417#31417 Answer by Jeff Strom for Question about the fundamental group and homotopy equivalence Jeff Strom 2010-07-11T14:39:17Z 2010-07-11T14:39:17Z <p>They are clearly not homotopy equivalent, and they have different homotopy groups. The nontorus is the cofiber of any injective map from a discrete $3$-point set to $S^2$, which is homotopy equivalent to $S^2\vee S^1\vee S^1$.</p>