# Topological manifolds

## Introduction

### Manifolds, charts and atlases

A manifold is a set of points and associated charts.

A chart is a mapping from each point in a subset of the manifold to a point in a vector space.

These charts are invertible. If we are given coordinates, we can identify the point in the manifold it comes from.

For each point we have a topological neighbourhood. For each point in the neighbourhood, we can map to an element in the tangent space.

#### Example: The sphere

We can map a hemisphere to a subset of $$R^2$$. Given a point in $$R^2$$ we can identify a specific point on the hemisphere, and given a s specific point on the hemisphere we can identify a point in $$R^2$$.

#### Universal charts

If the vector space is flat and non-repeating, then a single chart can be used to map the whole manifold.

#### Atlases

If we have a collection of charts which covers each point needs to be covered at least once, we have an atlas. Each chart needs to be to the same dimensional vector space.

### Transition maps

Where two charts overlap we can express the points where the charts overlap as two different coordinates.

We can express the mapping from these coordinates as a function. This is a transition map.

#### Overlapping charts

If two charts cover some of the same points on a manifold then we can define a function for those points where we move from one vector to another.

We can represent moving between charts as:

$$ab^{-1}$$

### Mapping 2D manifolds to Riemann surfaces

Needs to be orientable and metricisable.

### Connections of topological manifolds

Connected vs path-connected topological manifods.

## Paths

### Paths and loops

#### Paths

We have the set $$X$$. We define a mapping $$[0,1]\rightarrow X$$

If a path exists between any two points, then the space is path-connected.

#### Loops

This is a path which ends on itself.

If $$f(0)=f(1)$$ then it is a loop.

### Holes and genuses

#### Genes

The genus of a topology is the number of holes in the topology.

## Functions between topologies

### Functions between topologies

We can define a function from topology to another.

$$f(X)=Y$$

#### Continuous functions between topologies

If $$f(X)$$ is continous, then we have a continous function between topologies.

#### Inverse functions between topologies

If $$f(X)$$ is invertible then there is a inverse mapping.

### Homeomorphisms

If there is a mapping which is invertible and continuous, it is a homeomorphism.

## Fibre bundles

### Vector bundles

A vector bundle consists of a base manifold (a base space), and a real vector space at each point in the base manifold.

#### Example

For example we can have a base manifold of a circle, and have a $$1$$-dimensional vector space at each point on the circle to create an infinitely extended cylinder.

### Bundle projection

This is a projection from any point on any of the fibres, to the underlying base manifold.

### Trivial bundles and the torus

#### The torus

$$S_1 \times S_1$$

### Twisted bundles and the Klein bottle

#### Klein bottles

$$S_1 \times S_1$$, but twisted

### Mobius strips

$$S_1 \times$$ line segment.

## Other

### Submanifolds

Submanifold: subset of manifold which is also manifold

Eg: circle inside a sphere

### Boundries and interiors

Around every manifold of dimension $$n$$ is a boundry of dimension $$(n-1)$$.

Homeomorphism at boundry: one coordinate always $$\ge 0$$. reduced dimension.

Interior is rest.

### Embeddings and immersions

Whitney embedding theroem: all manfiolds can be embedded in $$R^n$$ space for some $$n$$.

### Topological groups

We have two operations for groups: multiplication and inversion.

A group is topological if these functions are continuous. + need to just read up on this. where is this relevant? + topological space

For these functions to be continous we need a metric defined on the group.