Equivariant topology

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 ${\displaystyle G}$ acts freely on ${\displaystyle X}$. Then, given a ${\displaystyle G}$-equivariant map ${\displaystyle f:X\to _{G}Y}$, we obtain sections ${\displaystyle s_{f}:X/G\to (X\times Y)/G}$ given by ${\displaystyle [x]\mapsto [x,f(x)]}$, where ${\displaystyle X\times Y}$ gets the diagonal action ${\displaystyle g(x,y)=(gx,gy)}$, and the bundle is ${\displaystyle p:(X\times Y)/G\to X/G}$, with fiber ${\displaystyle Y}$ and projection given by ${\displaystyle p([x,y])=[x]}$. Often, the total space is written ${\displaystyle X\times _{G}Y}$.

More generally, the assignment ${\displaystyle s_{f}}$ actually does not map to ${\displaystyle (X\times Y)/G}$ generally. Since ${\displaystyle f}$ is equivariant, if ${\displaystyle g\in G_{x}}$ (the isotropy subgroup), then by equivariance, we have that ${\displaystyle g\cdot f(x)=f(g\cdot x)=f(x)}$, so in fact ${\displaystyle f}$ will map to the collection of ${\displaystyle \{[x,y]\in (X\times Y)/G\mid G_{x}\subset G_{y}\}}$. In this case, one can replace the bundle by a homotopy quotient where ${\displaystyle G}$ acts freely and is bundle homotopic to the induced bundle on ${\displaystyle X}$ by ${\displaystyle f}$.

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 ${\displaystyle P}$, we define the configuration space, ${\displaystyle X}$, which parametrizes all associated solutions to the problem (such as points, lines, or arcs.) Additionally, we consider a test space ${\displaystyle Z\subset V}$ and a map ${\displaystyle f:X\to V}$ where ${\displaystyle p\in X}$ is a solution to a problem if and only if ${\displaystyle f(p)\in Z}$. Finally, it is usual to consider natural symmetries in a discrete problem by some group ${\displaystyle G}$ that acts on ${\displaystyle X}$ and ${\displaystyle V}$ so that ${\displaystyle f}$ is equivariant under these actions. The problem is solved if we can show the nonexistence of an equivariant map ${\displaystyle f:X\to V\setminus Z}$.

Obstructions to the existence of such maps are often formulated algebraically from the topological data of ${\displaystyle X}$ and ${\displaystyle V\setminus Z}$.[3] An archetypal example of such an obstruction can be derived having ${\displaystyle V}$ a vector space and ${\displaystyle Z=\{0\}}$. In this case, a nonvanishing map would also induce a nonvanishing section ${\displaystyle s_{f}:x\mapsto [x,f(x)]}$ from the discussion above, so ${\displaystyle \omega _{n}(X\times _{G}Y)}$, the top Stiefel–Whitney class would need to vanish.

• The identity map ${\displaystyle i:X\to X}$ will always be equivariant.
• If we let ${\displaystyle \mathbf {Z} _{2}}$ act antipodally on the unit circle, then ${\displaystyle z\mapsto z^{3}}$is equivariant, since it is an odd function.
• Any map ${\displaystyle h:X\to X/G}$ is equivariant when ${\displaystyle G}$ acts trivially on the quotient, since ${\displaystyle h(g\cdot x)=h(x)}$ for all ${\displaystyle x}$.

• Equivariant cohomology
• Equivariant stable homotopy theory
• G-spectrum