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Let X be a space (topological space, manifold, etc) and let the group G act continuously on X. What extra (homotopical, homological, cohomological, diffeomorphical etc) data can extracted from X when taking the group action into consideration as compared to when completely ignoring G? I would love to see some examples.

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$\def\R{\mathbb{R}} \def\SL{\text{SL}} \def\SO{\text{SO}}$Often you use the group action to study $G$ and not just to study $X$.

Here is an example: what does $\SL_2(\R)$ look like as a manifold? You can solve this by thinking of the group directly, but an easier way is to note that it acts transitively on the upper half plane by Mobius transformations. Since the stabilizer of $i$ is the circle group $\SO_2(\R)$, we get $$ \SL_2(\R)\sim \mathcal{H} \times S^1 \sim \R^2 \times S^1. $$ (not an isomorphism of Lie groups!)

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I believe it's more the other way around: a group action of $G$ on a space $X$ allows to construct a new space, the quotient $G\backslash X$ (of course the quotient space is "nice" only under some technical conditions).

Recognizing that a space $Y$ is actually realized as the orbit space $G\backslash X$ of some simpler space $X$ can help understand better the geometry and other features of $Y$.

For instance, realizing the real projective plane as the quotient of the sphere $S^2$ by the antipodal action of the group with two elements, makes evident its non-orientability and gives immediately a concrete realization of the non-trivial element in the fundamental group.

The simplest cases are possibly those of the circle $S^1=\Bbb R/\Bbb Z$ and the torus $T=\Bbb R^2/\Bbb Z^2$.

There is actually an important situation where the action of a group $G$ is helpful to gain information on $G$ (not $X$), namely when $X$ is a linear space and $G$ acts via linear transformations. This is the object of Representation Theory.

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Here is a general class of examples. Let $G$ be a connected, simply-connected complex semisimple group with a transitive algebraic action on a smooth complex projective variety $X$. Prior to thinking about the action of $G$ on X, there is little one can say about the topology of $X$. By taking into account the action of $G$, we find that $X$ must be $G$-equivariantly isomorphic to a coset variety of the form $G/P$, where $P$ is a parabolic subgroup of $G$. Accordingly, we may instead focus on the topology of $G/P$.

Choose a maximal torus $T$ and a Borel subgroup $B$ such that $T\subseteq B\subseteq P$. Let $W$ denote the Weyl group and $W_P$ the subgroup determined by $P$. We have the Bruhat cell decomposition $$G/P=\coprod_{[w]\in W/W_P}BwP/P,$$ where each Bruhat cell $BwP/P$ has dimension equal to twice the length of the corresponding minimal coset representative. In particular, the cells are even-dimensional. So, $X$ must be simply-connected, and $X$ has integral homology and cohomology only in even degrees.

For a concrete example, consider the action of $\SL_n(\mathbb{C})$ on the projective variety $X$ of full flags of subspaces of $\mathbb{C}^n$. It is not immediately clear that $X$ is simply-connected. However, once one sees that $\SL_n(\mathbb{C})$ acts transitively on $X$, one knows $X$ to be simply-connected.

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