
Real Analysis Real Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
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 induced by the inclusion map is injective and that is surjective with kernel . Thus, and by taking the abelianizations: 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 
Search tags for this page 
Click on a term to search for related topics.

Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
surface area  Kinroh  Physics  2  January 16th, 2014 04:14 PM 
compact subsurfaces of bordered surfaces of infinite genus  FrankD  Real Analysis  0  February 19th, 2013 02:43 AM 
[Help] Genus: Having difficulty to fully understand it.  probiner  Algebra  4  February 17th, 2012 11:49 AM 
Surface  mia6  Calculus  3  February 15th, 2010 06:54 PM 
torus, homeomorphic, deformation retraction  zelda2139  Real Analysis  0  February 19th, 2009 10:05 AM 