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On Fundamental Group - 2

Algebraic Topology
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Tsakanikas Nickos
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On Fundamental Group - 2

#1

Post by Tsakanikas Nickos » Tue Mar 15, 2016 4:51 pm

Let \( X \) be a topological space. Show that if \(a,b\) are points in \(X\) joined by a curve \(\gamma\), then the fundamental groups \(\pi_{1}(X,a)\) and \(\pi_{1}(X,b)\) are isomorphic. Note that this isomorphism in general depends on the curve \(\gamma\). However, show that if \(\pi_{1}(X,a)\) is abelian, then \(\pi_{1}(X,a)\) and \(\pi_{1}(X,b)\) are canonically isomorphic, i.e. the isomorphism does not depend on the curve joining \( a\) and \(b\).
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