In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the
classifying space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e., a topological space all of whose homotopy groups are trivial) by a proper free ...
for the
orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
O(''n'') may be constructed as the
Grassmannian
In mathematics, the Grassmannian \mathbf_k(V) (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all k-dimension (vector space), dimensional linear subspaces of an n-dimensional vector space V over a ...
of ''n''-planes in an infinite-dimensional real space
.
Cohomology ring
The
cohomology ring In mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomology' is usually un ...
of
with coefficients in the
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of
two elements is generated by the
Stiefel–Whitney classes:
[Hatcher 02, Theorem 4D.4.]
:
Infinite classifying space
The canonical inclusions
induce canonical inclusions
on their respective classifying spaces. Their respective colimits are denoted as:
:
:
is indeed the classifying space of
.
See also
*
Classifying space for U(''n'')
*
Classifying space for SO(n) In mathematics, the classifying space \operatorname(n) for thspecial orthogonal group'' \operatorname(n) is the base space of the universal \operatorname(n) principal bundle \operatorname(n)\rightarrow\operatorname(n). This means that \operatorname ...
*
Classifying space for SU(n) In mathematics, the classifying space \operatorname(n) for the special unitary group \operatorname(n) is the base space of the universal \operatorname(n) principal bundle \operatorname(n)\rightarrow\operatorname(n). This means that \operatorname(n) ...
Literature
*
*
*
External links
*
classifying space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e., a topological space all of whose homotopy groups are trivial) by a proper free ...
on
nLab
The ''n''Lab is a wiki for research-level notes, expositions and collaborative work, including original research, in mathematics, physics, and philosophy, with a focus on methods from type theory, category theory, and homotopy theory. The ''n''Lab ...
*
BO(n) on nLab
References
Algebraic topology
{{topology-stub