In mathematics, equivariant topology is the study of topological spaces that possess certain symmetries.
In studying topological spaces, one often considers continuous maps
, 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
, a property usually denoted by
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
necessarily vanishes.
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
acts freely on
gets the diagonal action
{\displaystyle g(x,y)=(gx,gy)}
Often, the total space is written
More generally, the assignment
(the isotropy subgroup), then by equivariance, we have that
In this case, one can replace the bundle by a homotopy quotient where
acts freely and is bundle homotopic to the induced bundle on
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.
[1][2] This is accomplished by using the configuration-space test-map paradigm: Given a geometric problem
, we define the configuration space,
, which parametrizes all associated solutions to the problem (such as points, lines, or arcs.)
Additionally, we consider a test space
is a solution to a problem if and only if
Finally, it is usual to consider natural symmetries in a discrete problem by some group
The problem is solved if we can show the nonexistence of an equivariant map
Obstructions to the existence of such maps are often formulated algebraically from the topological data of
[3] An archetypal example of such an obstruction can be derived having
In this case, a nonvanishing map would also induce a nonvanishing section
, the top Stiefel–Whitney class would need to vanish.