Classifying space for O(n)
Encyclopedia
In mathematics
, the classifying space
for O(n) may be constructed as the Grassmannian
of n-planes in an infinite-dimensional real Hilbert space
.
It is analogous to the classifying space for U(n)
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the classifying space
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 by a free action of G...
for O(n) may be constructed as the Grassmannian
Grassmannian
In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...
of n-planes in an infinite-dimensional real Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
.
It is analogous to the classifying space for U(n)
Classifying space for U(n)
In mathematics, the classifying space for the unitary group U is a space B together with a universal bundle E such that any hermitian bundle on a paracompact space X is the pull-back of E by a map X → B unique up to homotopy.This space with its universal fibration may be constructed as either# the...
.