TITLE: Why care about group actions?
QUESTION [11 upvotes]: 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.

REPLY [3 votes]: $\newcommand{\SL}{\operatorname{SL}}$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.