In mathematics, equivariant topology is the study of topological spaces that possess certain symmetries. In studying topological spaces, one often considers continuous function (topology), continuous maps $f:\; X\; \backslash to\; Y$, and while equivariant topology also considers such maps, there is the additional constraint that each map "respects symmetry" in both its domain of a function, domain and codomain, target space.
The notion of symmetry is usually captured by considering a Group action (mathematics), group action of a group (mathematics), group $G$ on $X$ and $Y$ and requiring that $f$ is Equivariant map, equivariant under this action, so that $f(g\backslash cdot\; x)\; =\; g\; \backslash cdot\; f(x)$ for all $x\; \backslash in\; X$, a property usually denoted by $f:\; X\; \backslash to\_\; Y$. 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 Borsukâ€“Ulam theorem, which asserts that every $\backslash mathbf\_2$-equivariant map $f:\; S^n\; \backslash to\; \backslash mathbb\; R^n$ necessarily vanishes.

Induced ''G''-bundles

An important construction used in equivariant cohomology and other applications includes a naturally occurring group bundle (see principal bundle for details). Let us first consider the case where $G$ acts Group_action#Types_of_actions, freely on $X$. Then, given a $G$-equivariant map $f:X\; \backslash to\_G\; Y$, we obtain sections $s\_f:\; X/G\; \backslash to\; (X\; \backslash times\; Y)/G$ given by $[x]\; \backslash mapsto\; [x,f(x)]$, where $X\; \backslash times\; Y$ gets the diagonal action $g(x,y)=(gx,gy)$, and the bundle is $p:\; (X\; \backslash times\; Y)/G\; \backslash to\; X/G$, with fiber $Y$ and projection given by $p([x,y])=[x]$. Often, the total space is written $X\; \backslash times\_G\; Y$. More generally, the assignment $s\_f$ actually does not map to $(X\; \backslash times\; Y)/G$ generally. Since $f$ is equivariant, if $g\; \backslash in\; G\_x$ (the isotropy subgroup), then by equivariance, we have that $g\; \backslash cdot\; f(x)=f(g\; \backslash cdot\; x)=f(x)$, so in fact $f$ will map to the collection of $\backslash $. In this case, one can replace the bundle by a Equivariant cohomology#Homotopy quotient, homotopy quotient where $G$ acts freely and is bundle homotopic to the induced bundle on $X$ by $f$.Applications to discrete geometry

In the same way that one can deduce the ham sandwich theorem from the Borsuk-Ulam Theorem, one can find many applications of equivariant topology to problems of discrete geometry. This is accomplished by using the configuration-space test-map paradigm: Given a geometric problem $P$, we define the ''configuration space'', $X$, which parametrizes all associated solutions to the problem (such as points, lines, or arcs.) Additionally, we consider a ''test space'' $Z\; \backslash subset\; V$ and a map $f:X\; \backslash to\; V$ where $p\; \backslash in\; X$ is a solution to a problem if and only if $f(p)\; \backslash in\; Z$. Finally, it is usual to consider natural symmetries in a discrete problem by some group $G$ that acts on $X$ and $V$ so that $f$ is equivariant under these actions. The problem is solved if we can show the nonexistence of an equivariant map $f:\; X\; \backslash to\; V\; \backslash setminus\; Z$. Obstructions to the existence of such maps are often formulated abstract algebra, algebraically from the topological data of $X$ and $V\; \backslash setminus\; Z$. An archetypal example of such an obstruction can be derived having $V$ a vector space and $Z\; =\; \backslash $. In this case, a nonvanishing map would also induce a nonvanishing section $s\_f:x\; \backslash mapsto\; [x,f(x)]$ from the discussion above, so $\backslash omega\_n(X\; \backslash times\_G\; Y)$, the top Stiefelâ€“Whitney class would need to vanish.Examples

* The identity map $i:X\; \backslash to\; X$ will always be equivariant. * If we let $\backslash mathbf\_2$ act antipodally on the unit circle, then $z\; \backslash mapsto\; z^3$is equivariant, since it is an Even and odd functions, odd function. * Any map $h:X\; \backslash to\; X/G$ is equivariant when $G$ acts trivially on the quotient, since $h(g\backslash cdot\; x)=h(x)$ for all $x$.See also

*Equivariant cohomology *Equivariant stable homotopy theory *G-spectrumReferences

{{Reflist Group actions (mathematics) Topological spaces Topology