Topology

There are various ways to build a topological space: As a subset of some bigger space; as the union of other spaces (glueing together boundaries); as the product of lower dimensinoal spaces; as the quotient of two spaces. We then want to determine when different constructions are essentially the same (i.e. can be smoothly transformed into each other).

Spheres

The 3-spheres can be defined as the set of points in \(\mathbb{R}^3 \) subject to a restriction. In other words the definition is a set of real numbers, together with the proof that these numbers satisify the equation of a sphere.
The 1-sphere (circle) can also be defined by \( S_1 = \{x,y \in \mathbb{R}| x + 1 \sim y \} \) or, equivalently, \(S_1 = \mathbb{R}/\mathbb{Z} \)

Subsets

We can use the conditional construct to form subsets:
So for a circle this is: We can use the conditional construct to form subsets:

Homeomorphism and Isomorphisms

Homeomophism and isomorphisms. Consider a torus embedded in 4-dimensions. It can be parameterised by \(\theta\) and \(\phi\).

Poincaré Conjecture

This relates to showing that a simply closed 3-manifold with trivial group is homeomorphic to the 3-sphere. (Homeomorphic is Greek for `same-shape') It is solved and proved to be true

Hodge Conjecture

More-or-less this states that certain aspects of the topology of a manifold defined by an algebraic equation in complex (projective) space can be defined in terms of other algebraic sub-manifolds.

CW-Complexes and Simplex Complexes

We could specify a topology by a list of vertices, edges, faces and so on.
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