Topology/Homotopy

Algebraic topology is the branch of topology where algebraic methods are used to solve topological problems. First, let's recall the fundamental problem of topology; given topological spaces $X$ and $Y$, to determine whether they are homeomorphic. Recall that two spaces are homeomorphic if and only if there exists a homeomorphism, that is, an open continuous bijection, between them. Thus, to conclude that two spaces are not homeomorphic, we need to go through each and every continuous map between them and check that it is not a homeomorphism! In general, this is impossible. Thus we need methods to deal with this problem. Algebraic topology makes some progress along these lines by assigning so-called algebraic invariants to topological spaces, in such a way that homeomorphic spaces have isomorphic invariants. Conversely, that means that if two spaces have different algebraic invariants, then they cannot be homeomorphic! Checking whether two algebraic structures are isomorphic or not in in general much easier than the original problem of homeomorphism, so this is a huge step forward.

Over the years, a multitude of different algebraic invariants have been developed. Whenever one designs or implements an invariant, it is important to have a good balance of computability versus completeness. We need to be able to compute the invariant invariant, and it must be "fine" enough to distinguish the properties we want to check. There is a fine line between vacuous computability and non-computable information! An invariant which achieves a good balance are the homotopy groups of a space, so we will start here. The homotopy groups are an infinite sequence $\pi _{0}(X),\pi _{1}(X),\pi _{2}(X),...$ of groups assigned to a space $X$. In this chapter, we will only concern ourselves with the first two groups, namely $\pi _{0}(X)$ and $\pi _{1}(X)$, as these are the easiest to compute. We will come back to the rest of the sequence in a while.

Paths and Loops

We have worked with the concept of paths before in the concepts of path connectedness and local path connectedness. Here, we will review them again, and then define some new terms.

Paths

Definition: We denote by $I$ the unit interval $[0,1]$ equipped with the subspace topology with respect to $\mathbb {R}$.

Definition: A path from $x$ to $y$ in a space $X$ is a continuous function from $I$ to $X$ such that $f(0)=x$ and such that $f(1)=y$.

Loops

Definition: Let $X$ be a topological space and $a\in X$. One says that $\alpha$ is a loop with base $a$ if $\alpha$ is path from a to a.

Definition

Let X and Y be topological spaces, and let f(x) and g(x) be continuous functions from X to Y. A homotopy between f and g is a continuous function h(x,r) from the set X×[0,1] to Y, such that h(x,0)=f(x), and such that h(x,1)=g(x).

Intuitively, we can think of a homotopy between two functions as a kind of continuous mapping between the two functions.

One can easily verify that homotopy is an equivalence relation both on paths and loops.

Homotopy of paths:
We define two paths to be homotopic when we consider a homotopy through paths through fixed starting an end points $x_{0}$ and $x_{1}$. The additional conditions are:

• $h(0,t)=x_{0}$ for any $t\in I$
• $h(1,t)=x_{1}$ for any $t\in I$

Homotopy of loops:
A homotopy of loops is the special case where the paths being considered is a loop, meaning with same starting and end point.

Note: If $X=\mathbb {R} ^{n}$ then all loops with base $a$ are homotopic. We just have to take

$F(t,s)=(1-s)\alpha (t)+s\beta (t).$

The same argument also goes through for any convex set.

• Theorem: path homotopy is an equivalence relation of paths.

Denote the path homotopy equivalence class of a path $f$ by $[f]$

Fundamental Groupoid

If $f$ is path from $x_{0}$ to $x_{1}$ and g is a path from $x_{1}$ to $x_{2}$, then define $f\star g$ as the path from $x_{0}$ to $x_{2}$ as follows:

$f\star g(t)={\begin{cases}f(2t){\text{ for }}t\in [0,{\frac {1}{2}}]\\g(2t-1){\text{ for }}t\in [{\frac {1}{2}},1]\end{cases}}$