My Math Forum  

Go Back   My Math Forum > College Math Forum > Real Analysis

Real Analysis Real Analysis Math Forum

LinkBack Thread Tools Display Modes
July 1st, 2011, 04:46 PM   #1
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'.
d'Artagnan is offline  

  My Math Forum > College Math Forum > Real Analysis

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

Copyright © 2019 My Math Forum. All rights reserved.