 #1
Math Amateur
Gold Member
 1,000
 41
Main Question or Discussion Point
I am reading Munkres book on Topology, Part II  Algegraic Topology Chapter 9 on the Fundamental Group.
On page 348 Munkres gives the following Lemma concerned with the homomorphism of fundamental groups induced by inclusions":
" Lemma 55.1. If A is a retract of X, then the homomorphism of fundamental groups induced by inclusion [itex] j: A \rightarrow X [/itex] is injective"
I am struggling with the proof  not so much intuitively  but in formulating a formal and explicit proof.
Because explaining my postion requires diagrams I have set out my problem in an attachment  see the attachment "Retractions and Induced Homomorphisms.
I have also provided an attachement of the relevant pages of Munkres book
I would like as much as anything a confirmation that my reasoning in the attachment "Problem ... ... " is correct. I would also be most interested to see how to formulate a formal and explicit proof of the Lemma
Peter
On page 348 Munkres gives the following Lemma concerned with the homomorphism of fundamental groups induced by inclusions":
" Lemma 55.1. If A is a retract of X, then the homomorphism of fundamental groups induced by inclusion [itex] j: A \rightarrow X [/itex] is injective"
I am struggling with the proof  not so much intuitively  but in formulating a formal and explicit proof.
Because explaining my postion requires diagrams I have set out my problem in an attachment  see the attachment "Retractions and Induced Homomorphisms.
I have also provided an attachement of the relevant pages of Munkres book
I would like as much as anything a confirmation that my reasoning in the attachment "Problem ... ... " is correct. I would also be most interested to see how to formulate a formal and explicit proof of the Lemma
Peter
Attachments

141.2 KB Views: 81

84.8 KB Views: 198