Maths - Algegraic Topology

Here we are concerned with shapes, when they are the same shape and the mappings between them.

For example, continuous mappings as allow a toroid to be continuously mapped into a mug.

It is like a sort of 'rubber geometry' where there are no absolute distances.

mapping

Determining Same Shape

There are different ways to 'equate' shapes such as homotopy equivalence and homeomorphisms. Homotopy equivalence tells us that a shape can be continuously deformed into another shape. A homeomorphism is a stricter comparison.

For instance a Möbius band is homotopy equivalent to a band but they are not homeomorphic. diagram

Homotopy Equivalence

Homotopy equivalence tells us that a shape can be continuously deformed into another shape. we can see that a Möbius band can be continuously deformed to a non-twisted band by first contracting it continuously to a circle and then continuously expanding it to a band.

diagram

How this can be defined more precisely?

One possibility might be to map every point in X to a point in Y and require an inverse map which brings the point back to where it started. duagram
diagram

However this doesn't work when, say, mapping a circle to a line. We can find ways to map every point on the circle to the line and back again but they are different shapes.

We need an extra requirement that every pair of points which are adjacent on X must be adjacent when mapped to Y. In this case there is at least one point where this is not true so this gives a way to show the circle and the line are different shapes.

So we need a way to make this continuity idea more precise.

Homeomorphism

In two dimensions it is relatively easy to determine if two spaces are topologically equivalent. We can check if they:

However, when we scale up to higher dimensions this does not work and it can become impossible to determine homeomorphism. There are methods which will, at least, allow us to prove more formally when topological objects are not homeomorphic.

These methods use 'invariants': properties of topological objects which do not change when going through a homeomorphism. Here we look at two types of invariants which arise from homotopy and homology.

Homotopy

It would take up too much space to properly explain the concept here so see this page for more details


nth homotopy group

If 2 spaces are homotopy equivilant the fundamental groups are isomorphic.

In homotopy we use equivalence classes between a circle and loops (which don't collapse) on the topological object that we are investigating. We can get further invariants by extending the circle to the surface of higher order n-spheres.

We can then get algebraic structures (mostly groups) by investigating what happens when these loops are composed, the loops are generators of the group. In homotopy the order of these compositions can be significant, that is the groups are not necessarily abelian.

Homology

It would take up too much space to properly explain the concept here so see this page for more details

homology groups

In homology we dont just use n-spheres but every closed oriented n-dimensional sub-manifold.

It also uses a different definition of equivalence classes where composition of loops commutes. This results in abelian groups.

So we dont need to fix the basepoint.

Example

Are the two shapes on the right 'homotopy equivalent' ?

Is there a continuous map between them in both directions?

There is more about this example on the page here.

diagram

Topology on a Set

A topology on a set X is a collection Τ of subsets of X, called open sets satisfying the following properties:

The requirements for the existence of of meets and joins correspond to the requirements for the existence of unions and intersections of open sets. Therefore these lattice structures can represent topologies.

  Lattice Open Set
  lat 1 open 1
  lat2 open 2
     
invalid - every intersection should be an open set: lat open
Venn Diagram Topological Space (or not) Lattice (frame)
topological space example 1 This is not a topological space because 'a' and 'b' are subsets but not the union of 'a' and 'b' lattice
topological space example 2 This is now a topological space because we have added the union of 'a' and 'b' lattice2
topological space example 3 This is not a topological space because 'ab' and 'bc' are subsets but not their intersection. lattice3
topological space example 4 This is now a topological space because we have added 'b' lattice4

Note: Assume Ø is included in the above examples.

Topological Space

In many cases the concept of a metric space is unnecessary, however we still need the concept of 'nearness' and hence 'continuity'. 'Topological space' based on the 'topological open set' is the most general way we can do this. This allows us to define nearness purely using the concept of a subset.

Hausdorff Space

Hausdorff space is a bit more specific than general topological space. Space is Hausdorff if, in addition to being a topological open set, for any two points:
x1, x2∈X there are disjoint open sets U1, U2 that contain x1and x2.

Basis

A basis is a subcollection Bcontains= U

where

such that evey element of U is a union of open sets in B.

Link between Topology and Logic

One way to visualise the link between topology and logic is to start with a Venn diagram.

We can then map points in the Venn diagram to either true or false depending on whether they are in a given set.

topology and logic

This is a nice way to link geometry, logic and topology.

see pages about:

Link between Topology and Category Theory

Presheaf

 

Fibre in Category Theory

See page here.

Slice Category

See page here.

Examples

Non-Geometric Examples

Sequences

Here the concept of 'nearness' comes from sequences that begin in the same way.

Denotional Semantics

 

Recursively Defined Structures

 

Further Information

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