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A $G$-complex is a CW-complex $X$ together with an action of $G$ on $X$ which permutes the cells (I think the action should be continuous). This action induces an action on the cellular chain complex $C_*(X)$. My question is: why the differential is a map of G-modules?

i.e.: why $\partial(ge^\alpha_n)=g\partial(e^\alpha_n)$?

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    Notice that, with your definition, the map $m_g: X \rightarrow X$ is a cellular map, and so induces a *map of chain complexes* $(m_g)_*: C_*(X) \rightarrow C_*(X)$. By definition, maps of chain complexes commute with the boundary map. So this is really just a consequence of the more general fact that cellular maps induce chain maps on the cellular chain complex.2011-09-03
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    Are we using continuity somewhere?2011-09-03
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    cellular ---> continuous by the definition of the topology of the complex2011-09-03
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    I meant the continuity of the action2011-09-03
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    At least in Hatcher and every time I have seen this the continuity of cellular maps is usually implied (the book/chapter says all maps are continuous in the beginning or something like it somewhere) but not explicitly said.2011-09-06

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