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.
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 \(R^2\).
If the vector space is flat and non-repeating, then a single chart can be used to map the whole manifold.
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.
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.
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:
Needs to be orientable and metricisable.
Connected vs path-connected topological manifods.
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.
This is a path which ends on itself.
If \(f(0)=f(1)\) then it is a loop.
The genus of a topology is the number of holes in the topology.
We can define a function from topology to another.
If \(f(X)\) is continous, then we have a continous function between topologies.
If \(f(X)\) is invertible then there is a inverse mapping.
If there is a mapping which is invertible and continuous, it is a homeomorphism.
A vector bundle consists of a base manifold (a base space), and a real vector space at each point in the base manifold.
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.
This is a projection from any point on any of the fibres, to the underlying base manifold.
\(S_1 \times S_1\)
\(S_1 \times S_1\), but twisted
\(S_1 \times \) line segment.
Submanifold: subset of manifold which is also manifold
Eg: circle inside a sphere
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.
Whitney embedding theroem: all manfiolds can be embedded in \(R^n\) space for some \(n\).
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.