nLab hyperplane




Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




By a hyperplane one usually means an affine or linear subspace of an affine space or linear space, respectively, typically required to be a positive dimension and codimension and often required to have codimension = 1.


In the archetypical example the ambient space is the time-honored Euclidean/Cartesian space 3\mathbb{R}^3 and a hyperplane of codimension 1 is an ordinary plane.

More sophisticated examples arise for instance in the discussion of configuration spaces of points in a Cartesian space E= dE = \mathbb{R}^d, where hyperplanes in E nE^n given by {(x 1,,x n)E n|x i=x j}\big\{ (\vec x_1, \cdots, \vec x_n) \,\in\, E^n \;\big\vert\; \vec x_i = \vec x_j \big\} reflect the subspaces where positions of a given pair of points coincide. This leads to discussion of arrangements of hyperplanes and their complements, see also at Knizhnik-Zamolodchikov equation the references on twisted cohomology of configuration spaces of points.


See also:

Created on July 14, 2022 at 11:28:09. See the history of this page for a list of all contributions to it.