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.

Dimension theory

Refinement

Ply (order) of a cover

Small inductive dimension

Large inductive dimension

Lebesgue covering dimension

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

Holes

Genes

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

Path-connect spaces

Simply-connected 2D manifolds

Elliptic (Riemann sphere)

Parabolic (complex plane)

Hyperbolic (open disk)

Not simply-connected 2D manifolds

Torus

Hyper-elliptic curves

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.

Homotopy

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 and twisted bundles

Cross-sections and zero-sections of fibre bundles

Trivial bundles and the torus

Trivial bundles

The torus

\(S_1 \times S_1\)

Twisted bundles and the Klein bottle

Twisted bundles

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.