Equivariant topology explained
, 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.
on
and
and requiring that
is
equivariant under this action, so that
for all
, 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
-equivariant map
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
acts freely on
. Then, given a
-equivariant map
, we obtain sections
given by
, where
gets the diagonal action
, and the bundle is
, with fiber
and projection given by
. Often, the total space is written
.
More generally, the assignment
actually does not map to
generally. Since
is equivariant, if
(the isotropy subgroup), then by equivariance, we have that
, so in fact
will map to the collection of
\{[x,y]\in(X x Y)/G\midGx\subsetGy\}
. In this case, one can replace the bundle by a homotopy quotient where
acts freely and is bundle homotopic to the induced bundle on
by
.
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.[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
and a map
where
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
that acts on
and
so that
is equivariant under these actions. 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
and
.
[3] An archetypal example of such an obstruction can be derived having
a
vector space and
. In this case, a nonvanishing map would also induce a nonvanishing section
from the discussion above, so
, the top
Stiefel–Whitney class would need to vanish.
Examples
will always be equivariant.
act antipodally on the unit circle, then
is equivariant, since it is an
odd function.
is equivariant when
acts trivially on the quotient, since
for all
.
See also
Notes and References
- Book: Matoušek, Jiří. Jiří Matoušek (mathematician)
. Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry. Using the Borsuk–Ulam Theorem. Jiří Matoušek (mathematician). Universitext. Springer. 2003.
- Book: Handbook of Discrete and Computational Geometry, Second Edition. 2004-04-15. Chapman and Hall/CRC. 9781584883012. Goodman. Jacob E.. 2nd. Boca Raton. English. O'Rourke. Joseph.
- Web site: Equivariant topology methods In discrete geometry. Matschke. Benjamin.