User:Fropuff/Drafts/Miscellaneous

Miscellaneous drafts. To be merged with their respective articles when complete.

Topological manifold

Additional structure

Topological manifolds are much more useful often more tractable when given some additional structure. Much of the study of topological manifolds is, therefore, devoted to understanding conditions under which such structures exist and are unique.

Euclidean space

Affine structure

To study Euclidean geometry one does not really need to know the location of the origin in Rⁿ, any point is just as good as any other. This leads to a construction in mathematics known the affine space underlying any given vector space.

Group-theoretic perspective

• General linear group
• Orthogonal group
• Affine group
• Euclidean group

CW complex

A closed cell is a topological space homeomorphic to a ball (a sphere plus interior), or equally to a simplex, or a cube in n dimensions. Only the topological nature matters: but one does want to keep track of the subspace on the 'surface' (the sphere that bounds the ball), and its complement, the interior points. An open cell is the interior of a closed cell.

CW complexes are defined inductively by gluing together cells of successively higher dimensions. The complex constructed at the n^th stage is called the n-skeleton. One proceeds as follows:

1. Start with a discrete set X⁰ of 0-cells (i.e. points). This is the 0-skeleton.
2. Inductively glue a collection of (n+1)-dimensional cells to the n-skeleton Xⁿ via attaching maps, i.e. continuous maps f : ∂Dⁿ⁺¹ = Sⁿ → Xⁿ. The (n+1)-skeleton Xⁿ⁺¹ is defined as the quotient of the disjoint union of Xⁿ with the (n+1)-cells via the identifications made by the attacting maps (i.e. x ∼ f(x)).
3. Let X = ∪_nXⁿ equipped with the weak topology: a subset A ⊂ X is open iff A ∩ Xⁿ is open in Xⁿ for each n.

Pseudoscalar

The unit pseudoscalar in Cℓ_p,q(R) is given by

$\omega =e_{1}e_{2}\cdots e_{n}$

The norm of ω is given by

$Q(\omega )={\bar {\omega }}\omega =(-1)^{q}$

and the square is

$\omega ^{2}=(-1)^{n(n+1)/2}(-1)^{q}=(-1)^{(p-q)(p-q+1)/2}={\begin{cases}+1&p-q\equiv 0,3\mod {4}\\-1&p-q\equiv 1,2\mod {4}\end{cases}}$

List of small groups

Order 16

There are 14 groups (5 abelian) of order 16.

Names:	Dihedral group $D_{8}$
Description:	Symmetry group of an octagon. Semidirect product of $C_{8}$ by $C_{2}$.
Properties:
Presentation:	$\langle r,f\mid r^{8},f^{2},frf=r^{-1}\rangle$
Center:	$C_{2}$