User:Fropuff/Draft 1

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

Contents

[edit] Topological manifold

[edit] 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.

[edit] Euclidean space

[edit] 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.

[edit] Group-theoretic perspective

[edit] 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 nth 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.

[edit] Pseudoscalar

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

\omega = e_1e_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}

[edit] List of small groups

[edit] Order 16

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

Names: Dihedral group D8
Description: Symmetry group of an octagon. Semidirect product of C8 by C2.
Properties:
Presentation: \langle r, f \mid r^8, f^2, frf = r^{-1} \rangle
Center: C2