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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, equivariant topology is the study of
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
s that possess certain symmetries. In studying topological spaces, one often considers continuous maps 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 Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
and
target Target may refer to: Physical items * Shooting target, used in marksmanship training and various shooting sports ** Bullseye (target), the goal one for which one aims in many of these sports ** Aiming point, in field artillery, fi ...
space. The notion of symmetry is usually captured by considering a
group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
of a group G on X and Y and requiring that f is equivariant under this action, so that f(g\cdot x) = g \cdot f(x) for all x \in X, a property usually denoted by f: X \to_ Y. Heuristically speaking, standard
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
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 \mathbf_2-equivariant map f: S^n \to \mathbb R^n necessarily vanishes.


Induced ''G''-bundles

An important construction used in
equivariant cohomology In mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from algebraic topology which applies to topological spaces with a ''group action''. It can be viewed as a common generalization of group cohomology and an ordi ...
and other applications includes a naturally occurring group bundle (see
principal bundle In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
for details). Let us first consider the case where G acts freely on X. Then, given a G-equivariant map f:X \to_G Y, we obtain sections s_f: X/G \to (X \times Y)/G given by \mapsto ,f(x)/math>, where X \times Y gets the diagonal action g(x,y)=(gx,gy), and the bundle is p: (X \times Y)/G \to X/G, with fiber Y and projection given by p( ,y= /math>. Often, the total space is written X \times_G Y. More generally, the assignment s_f actually does not map to (X \times Y)/G generally. Since f is equivariant, if g \in G_x (the isotropy subgroup), then by equivariance, we have that g \cdot f(x)=f(g \cdot x)=f(x), so in fact f will map to the collection of \. In this case, one can replace the bundle by a 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 \subset V and a map f:X \to V where p \in X is a solution to a problem if and only if f(p) \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 \to V \setminus Z. Obstructions to the existence of such maps are often formulated algebraically from the topological data of X and V \setminus Z. An archetypal example of such an obstruction can be derived having V a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
and Z = \. In this case, a nonvanishing map would also induce a nonvanishing section s_f:x \mapsto ,f(x)/math> from the discussion above, so \omega_n(X \times_G Y), the top Stiefel–Whitney class would need to vanish.


Examples

* The identity map i:X \to X will always be equivariant. * If we let \mathbf_2 act antipodally on the unit circle, then z \mapsto z^3is equivariant, since it is an
odd function In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power se ...
. * Any map h:X \to X/G is equivariant when G acts trivially on the quotient, since h(g\cdot x)=h(x) for all x.


See also

*
Equivariant cohomology In mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from algebraic topology which applies to topological spaces with a ''group action''. It can be viewed as a common generalization of group cohomology and an ordi ...
* Equivariant stable homotopy theory * G-spectrum


References

{{Reflist Group actions (mathematics) Topological spaces Topology