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I got a little bit confused with Hatcher's explanation of fundamental group action on the chain complex of the universal cover (Algebraic Topology, A. Hatcher, p. 328).

Assume $X$ is a path-connected space with universal cover $\tilde{X}$. As I unterstood, there supposed to be a right action $*$ of $\pi_1(X)$ on $C_n(\tilde{X})$ deinfed for $[\gamma]\in\pi_1(X)$ and an $n$-singular simplex $\sigma\in C_n(\tilde{X})$ as follows $$ \sigma*[\gamma]=\tilde{\gamma}\circ\sigma $$ where $\tilde{\gamma}:I\rightarrow\tilde{X}$ is a unique lift of the loop $\gamma:I\rightarrow X$ with $p(\tilde{\gamma}(0))=\gamma(0)=\gamma(1)$ (here $p$ is the covering map).

My problem pops out when it comes to verifying group action properties. Namely we should have $\sigma*([\gamma_1][\gamma_2])=(\sigma*[\gamma_1])*[\gamma_2]$. But (denoting by $\centerdot$ the path concatenation) $$ LHS=\sigma*[\gamma_1\centerdot\gamma_2]=\widetilde{\gamma_1\centerdot\gamma_2}\circ\sigma=(\widetilde{\gamma_1}\centerdot\widetilde{\gamma_2})\circ\sigma $$ whereas $$ RHS=(\widetilde{\gamma_1}\circ\sigma)*[\gamma_2]=\widetilde{\gamma_2}\circ\widetilde{\gamma_1}\circ\sigma $$ I don't see how $LHS=RHS$. Should it be the case that $\widetilde{\gamma_1}\centerdot\widetilde{\gamma_2}=\widetilde{\gamma_2}\circ\widetilde{\gamma_1}$? It seems strange to me.

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  • $\begingroup$ $\gamma : X\to X$ is not even well-typed. However, for $\gamma : I\to X$ and a fixed $c\in p^{-1}(\gamma(0))$, there is a unique lift $\tilde{\gamma}: I\to \tilde{X}$ with $\tilde{\gamma}(0) = c$. Now $\tilde{\gamma}(1)\in p^{-1}(\gamma(0))$ ($\gamma$ is a loop), and so there is a unique automorphism $f$ of $(\tilde{X},p)$ such that $f(c)= \tilde{\gamma}(1)$. Then all the desired properties should come from the uniqueness at each stage $\endgroup$ – Max Dec 18 '18 at 17:29
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    $\begingroup$ I agree with @Max, and will add the comment that although the notation of that passage of Hatcher is highly abbreviated, part of your job in understanding that passage is to go back to the actual definitions and theorems regarding all objects under discussion and to use those definitions correctly. For example, your question exhibits several mistakes about the definitions and properties of the left action by deck transformations of the fundamental group of $X$ acting on the universal covering space of $X$ (described in Proposition 1.39 of Hatcher). $\endgroup$ – Lee Mosher Dec 19 '18 at 13:41

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