Equivariant topology

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In mathematics, equivariant topology is the study of topological spaces that possess certain symmetries. In studying topological spaces, one often considers continuous maps ${\displaystyle f:X\to Y}$, and while equivariant topology also considers such maps, there is the additional constraint that each map "respects symmetry" in both its domain and target space.

The notion of symmetry is usually captured by considering a group action of a group ${\displaystyle G}$ on ${\displaystyle X}$ and ${\displaystyle Y}$ and requiring that ${\displaystyle f}$ is equivariant under this action, so that ${\displaystyle f(g\cdot x)=g\cdot f(x)}$ for all ${\displaystyle x\in X}$, a property usually denoted by ${\displaystyle f:X\to _{G}Y}$. Heuristically speaking, standard topology views two spaces as equivalent "up to deformation," while equivariant topology considers spaces equivalent up to deformation so long as it pays attention to any symmetry possessed by both spaces. A famous theorem of equivariant topology is the Borsuk–Ulam theorem, which asserts that every ${\displaystyle \mathbf {Z} _{2}}$-equivariant map ${\displaystyle f:S^{n}\to \mathbb {R} ^{n}}$ necessarily vanishes.