Topology/Homotopy

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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 and , 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 btween 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 of groups assigned to a space . In this chapter, we will only concern ourselves with the first two groups, namely and , as these are the easiest to compute. We will come back to the rest of the sequence in a while.

Contents

Paths and LoopsEdit

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.

PathsEdit

Definition: We denote by   the unit interval   equipped with the subspace topology with respect to  .

Definition: A path from   to   in a space   is a continuous function from   to   such that   and such that  .

LoopsEdit

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

DefinitionEdit

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 loops:
We can define two loops to be homotopic, when we consider a homotopy through loops, i. e. a homotopy h(x,r) between the two loops such that h(x,0)=h(x,1) for all x.

Note: If   then all loops with base   are homotopic. We just have to take

 


Topology
 ← Deformation Retract Homotopy The fundamental group →