Algebraic Topology
In this post I will give an overview of the field of mathematics known as algebraic topology. In algebraic topology various algebraic objects are assigned to topological spaces which are invariant under homotopy equivalence. The hope is that these algebraic objects will allow us to distinguish between the underlying topological spaces.
Euler Characteristic
In the beginning of algebraic topology the algebraic objects assigned to topological spaces were of a simple nature. One of the first such invariants is called the Euler characteristic. Here the topological space is given a triangulation so it has a certain number of vertices, edges and faces which we will denote by V, E and F respectively. The Euler characteristic is then defined to be V - E + F, which is just an integer.
Whenever mathematicians discover a new invariant for topological spaces it is important to check that the answer does not depend on the method for choosing the invariant. For example, the Euler characteristic seems to depend on how we triangulate our topological space because we could triangulate the same space in multiple ways. This is true but it is a theorem that the quantity V - E + F will not change for two different triangulations of the same space. Thus the Euler characteristic gives us a well defined invariant of a topological space which is easy to compute.
Homotopy Groups
As algebraic topology progressed the invariants defined on the spaces become more complicated. Another type of invariant that can be assigned to a topological space is called a homotopy group. We can define a homotopy group for each dimension but we will start in dimension one with the first homotopy group called the fundamental group.
To define the fundamental group of a topological space we start with a basepoint in our space and consider the set of all maps from the unit interval in to our space which begin and end at the basepoint. Such a map is called a loop. When then define two loops to be equivalent if there is a homotopy between them.
The set of equivalence classes of loops in our topological space is called the fundamental group. The fundamental group forms an abstract group under multiplication of two loops given by traversing both loops at twice the speed as the original loops. This multiplication is associative with identity element the constant loop and the inverse of a loop is the same loop traversed backwards.
The fundamental group of a topological space is not always easy to calculate. Even for a simple space such as the circle this requires a calculation. However, it is well known that the fundamental group of the circle is the group of integers. Whereas for a disk the fundamental group is the trivial group. This is because any loop in a disk is homotopy equivalent to a constant loop.
To construct the higher dimensional homotopy groups we use a similar method as for the fundamental group. For example, the nth homotopy group is the equivalence classes of maps from the n sphere in to our topological space. It turns out that if n is greater than 1 then the nth homotopy group is abelian.
Calculating the higher dimensional homotopy groups is an extremely difficult problem in algebraic topology and is still an active area of research in mathematics today. Even for such simple manifolds as the n sphere certain homotopy groups are known but a complete solution has not been found. The machinery used to answer these questions is extremely complex and I will not attempt to explain it here.
Homology Groups
Another important set of invariants that can be assigned to a topological space are called homology groups. There are several equivalent ways of defining homology groups on a space that give isomorphic groups but all of the definitions require a bit of machinery. I will not go over this machinery but instead give some geometric intuition of what these groups measure.
Just as with the homotopy groups we have a homology group in every dimension. The homology group in dimension n measures any n dimensional holes that occur in our topological space. Thus even though we are dealing with abstract groups there is a geometric understanding behind them.
Homology groups are usually easier to calculate than homotopy groups and they are always abelian groups unlike the fundamental group. One of the important tools in understanding homology groups is called an exact sequence and many different exact sequences have popped up in various branches of mathematics after their importance was realized in algebraic topology.
Summary
In this post I have gone over three important invariants of a topological space which are studied in algebraic topology. There are many other invariants that mathematicians use which are too complex too describe in this post such as those that occur in the area of K-Theory. Algebraic topology is a branch of mathematics that has been around for over a century and will continue to thrive in the 21st century.
References:
https://en.wikipedia.org/wiki/Algebraic_topology
http://mathworld.wolfram.com/AlgebraicTopology.html
In an algebraic topological setting you would use singular homology to define the Euler characteristic. Hence, you need homology groups to define it (in its most general setting). So I think it makes more sense to discuss it last.
You can of course also define the Euler characteristic in a weaker setting by just considering simplicial complexes. This then reduces to the setting of polyhedral surfaces. However, if you use this definition it becomes a bit more restrictive.
So I guess it makes more sense to discuss how homotopy and homology connect to the Euler characteristic then the other way round.
The Euler characteristic was originally defined without the use of homology and I give it as a simple example here because most people do not know what homology is. Of course it can be defined using the Betti numbers or ranks of the homology but this came later.
Then you should say "that the Euler characteristic gives us a well defined invariant of a polyhedral surface which is easy to compute" instead of "that the Euler characteristic gives us a well defined invariant of a topological space which is easy to compute" since for a topological space you require a more general version of the Euler characteristic.