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I am reading J.P. May's book on "A Concise Course in Algebraic Topology" and have approached the calculation where [itex]\pi[/itex][itex]_{1}[/itex](S[itex]^{1})[/itex][itex]\cong[/itex]Z

He defines a loop f[itex]_{n}[/itex] by e[itex]^{2\pi ins}[/itex]

I want to show that [f[itex]_{n}[/itex]][f[itex]_{m}[/itex]]=[f[itex]_{m+n}[/itex]]

I understand this as trying to find a homotopy between f[itex]_{n}[/itex]*f[itex]_{m}[/itex] and f[itex]_{m+n}[/itex]

I have some attempts some attempts which have been unsuccessful are

H(s,t)= f[itex]_{n+mt}[/itex]*f[itex]_{m(1-t)}[/itex]

H(s,t)={e[itex]^{2\pi in2st}[/itex]e[itex]^{2\pi im2s(1-t)}[/itex] for s in [0,1/2]

{e[itex]^{2\pi im(2s-1)t}[/itex]e[itex]^{2\pi in(2s-1)(1-t)}[/itex] for s in [1/2,1]

Any help would be very much appreciated on my part.

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# Showing the Fundamental Group of S^1 is isomorphic to the integers

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