Equivariant topology

In mathematics, equivariant topology is the study of topological spaces that possess certain symmetries. In studying topological spaces, one often considers continuous maps formula_1, 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 formula_2 on formula_3 and formula_4 and requiring that formula_5 is equivariant under this action, so that formula_6 for all formula_7, a property usually denoted by formula_8. 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 formula_9-equivariant map formula_10 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 formula_2 acts freely on formula_3. Then, given a formula_2-equivariant map formula_14, we obtain sections formula_15 given by formula_16, where formula_17 gets the diagonal action formula_18, and the bundle is formula_19, with fiber formula_4 and projection given by formula_21. Often, the total space is written formula_22.
More generally, the assignment formula_23 actually does not map to formula_24 generally. Since formula_5 is equivariant, if formula_26 (the isotropy subgroup), then by equivariance, we have that formula_27, so in fact formula_5 will map to the collection of formula_29. In this case, one can replace the bundle by a homotopy quotient where formula_2 acts freely and is bundle homotopic to the induced bundle on formula_3 by formula_5.
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 formula_33, we define the "configuration space", formula_3, which parametrizes all associated solutions to the problem (such as points, lines, or arcs.) Additionally, we consider a "test space" formula_35 and a map formula_36 where formula_37 is a solution to a problem if and only if formula_38. Finally, it is usual to consider natural symmetries in a discrete problem by some group formula_2 that acts on formula_3 and formula_41 so that formula_5 is equivariant under these actions. The problem is solved if we can show the nonexistence of an equivariant map formula_43.
Obstructions to the existence of such maps are often formulated algebraically from the topological data of formula_3 and formula_45. An archetypal example of such an obstruction can be derived having formula_41 a vector space and formula_47. In this case, a nonvanishing map would also induce a nonvanishing section formula_48 from the discussion above, so formula_49, the top Stiefel–Whitney class would need to vanish.