My Math Forum retraction on a surface of genus g

 Real Analysis Real Analysis Math Forum

 July 1st, 2011, 04:46 PM #1 Newbie   Joined: Aug 2010 Posts: 1 Thanks: 0 retraction on a surface of genus g Problem: In the surface Mg of genus g, let C be a circle that separates Mh' and Mk' obtained from the closed surfaces Mh and Mk by deleting an open disk from each. Show that Mh' does not retract onto its boundary circle C, and hence Mg does not retract onto C. Hatcher Allen. Algebraic Topology Section 1.2 Problem 9 My attempt: Suppose there was such a retraction. Then we would have that $i_*:\pi_1(C)\to\pi_1(M'_h)$ induced by the inclusion map is injective and that $\phi:\pi_1(M'_h)\to\pi_1(M_h)$ is surjective with kernel $i_*(\pi_1(C))$. Thus, $\pi_1(M'_h)/i_*(\pi_1(C))\approx \pi_1(M_h)$ and by taking the abelianizations: $\mathbb{Z}^{2h}/\mathbb{Z}\approx\mathbb{Z}^{2h}$ yielding a contradiction. Is this correct? I used the assumption that C was a retract of Mh' to say that the fundamental group of C is isomorphic to a subgroup of the fundamental group of Mh'.

 Tags genus, retraction, surface

### does not retract onto its boundary circle

Click on a term to search for related topics.
 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post Kinroh Physics 2 January 16th, 2014 04:14 PM FrankD Real Analysis 0 February 19th, 2013 02:43 AM probiner Algebra 4 February 17th, 2012 11:49 AM mia6 Calculus 3 February 15th, 2010 06:54 PM zelda2139 Real Analysis 0 February 19th, 2009 10:05 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top