First we must choose a finite set of vertices for our graph . Furthermore, we must introduce the notion of distance between these vertices. The most convenient way of do this is to embed into via a mapping . This embedding associates each to some point . This allows us to utilize the Euclidean metric:

We now impose a condition on our graph :

(i.) Given the edge set of , the probability that is dependent on .

(ii.) The probability of an edge belonging to is independent of all other edges.

We now impose conditions on :

(i.)

(ii.)

(iii.)

Clearly the function satisfies these conditions. Since and , we know .

We now define the separation, , of and as the length of the minimal path between and . Clearly and given a disconnected vertice , we define to be immeasurable. We also define the degree of separation of , as the average value of , as and range over all vertices. We know that for all vertices. We can now pose the following questions:

(i.) Given any number , what embedding , if any, will ensure ? Moreover, is this embedding necessarily unique? If not, then does the condition induce an equivalence class on such embeddings?

(ii.) Does there exist a algorithm to express given ? If so, what is its computational complexity?

In the next post, we will begin to explore these questions in great detail.

In general, the fundemental group is determined by the choice of basepoint. However, we can easily show (and it is quite intuitive) that for any path connected space (i.e. a space such that there exists a path between any two points), the fundemental groups at two unequal points in are isomorphic (they are equal). This fact can be applied to the arguments in the last post about the fundemental groups of euclidean spaces, as well as the circle, since they are path connected.

In the last post, I stated that the fundemental group acts as a functor from the category to the category . This means that for any continuous map , every loop in with basepoint is mapped to a loop in with basepoint where . This results in a group homomorphism called the induced homomorphism. Also, given any two continuous maps and where and (homotopic relative to ), then . This means that the functor does not distinguish between homotopic maps at a given bsepoint. Therefore, homotopy equivalent path connected spaces have isomorphic fundemental groups.

I stated that I we would prove a well know result using fundemental groups in this post. But I need something to talk about next time, so it will wait.

Now that we have motivated the topic at hand, we can begin. Since this is algebraic topology, I will assume you are already comfortable with some basic point-set topology. If not, I recommend that you search for “topology notes” on google. You’re bound to find a decent reference.

Earlier I spoke about topology being the study of certain objects under continuous deformations. These objects are called topological spaces and continuous deformations are simply continuous maps or functions. Remember, we call a map from a topological space to a topological space if takes open sets in to open sets in . For those who study category theory, continuous maps are the morphisms of the category .

In algebraic topology, and especially homotopy theory, we study objects on topological spaces called paths. We will find that paths allow us to define an extremely important construction called the fundemental group. A path is a continuous function where is the unit interval, , and is a topological space. We say a is a path from to if and . We call a path a loop if . Given two paths from to on a topological space that share endpoints, we define a path homotopy as a continuous map that has the following properties: , , , . Intuitively, this can be viewed as a continuous deformation of the original path to the second one. Also, it is correct to say that homotopy essentially counts the holes in a space.

It is easy to show that path homotopy is an equivalence relation. Therefore, given the set of all loops based at a point in a topological space , we place homotopic loops at into equivalence classes. In other words, all the loops at that have a homotopy between then (can be continuously deformed into each other) are put into a single class. We can denote a class of homotopic loops at as . If we consider the set of all eqivalence classes of loops based at a single point in a pointed topological space and the operation of concatenation (fancy word for connecting) we obtain a group called the fundemental group, . For example, imagine a topological space and point . Imagine that there are three equivalence classes of loops at (i.e. three sets homotopic of loops). Call them , , and . Then we define the concatenation of loops as (i.e. concatenation is well-defined). Do note that I am being quite liberal by defining the third equivalence class in this example as the concatenation of the others, but I’m simply trying to reinforce that this is a group.

Now for a concrete example of the fundamental group. Consider any Euclidean space . Then the fundemental group of at any basepoint contains only one element. This is due to the fact that all loops in can be deformed into each other (i.e. they are homotopic). Intuitively this is because there are no “holes” in a Euclidean space. Therefore the fundemental group (i.e. it is the trivial group of one element).

Another great example is that of the circle . Essentially, we consider loops that wrap around the circle times. We consider loops that move counter-clockwise as positive and those that move clockwise as negative. It is obvious that loops with the same number of turns are homotopic. Therefore, we get a one-to-one correspondence between the homotopy classes of and the integers. Therefore we can conclude that . Despite the simple reasoning, we will need a bit more machinery to actually prove this.

Now that we have established the fundemental group as a concrete construction, we examine its properties when acted on by continuous deformations (this is topology after all). Without getting too far into the next post, a homeomorphism of topological spaces induce homomorphism between the underlying fundemental groups.

Next time, we will discuss the implications of the fundemental group in a categorical sense (we will show that the fundemental group is a funtor from the catergory to ). We will also prove a very well known result using the fundemental group.