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Algebraic Topology - Retractions and Homomorpisms Induced by Inclusions

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
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 [tex] j: A \rightarrow X [/tex] 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