The Homotopy Invariance Theorem

Homotopy of Paths

Recall the definition of a homotopy:

A homotopy is a continuous deformation of one function into another. Formally, given two continuous functions , where are topological spaces, a homotopy between and is a continuous function such that and . One says that two maps and are homotopic, if there exists a homotopy from to , and denoted .

Link to original

Let be a domain, and be two paths in . So a homotopy from to is a continuous map such that and for all .

Closed Path Homotopy, Path with Constant Endpoints Homotopy

Homotopy is called a CP-homotopy if each is a closed path. i.e., for all . is called a PCE-homotopy if the beginning and end points of the paths are independent of . i.e. and for all .

e.g. The paths and are CP-homotopic in , as is a homotopy between them.

Lemma

Every closed path in is CP-homotopic in to one of the form , for .

The Homotopy Invariance Theorem

Lemma

Let , with compact and open. Then there exists such that for all , .

Proof Assume the contrary. Then for all , there is some such that . Let , a closed set. Choose sequence , such that . Since is compact, there is a subsequence converging to some . Then . Since is closed, , thus , contradicting .

Homotopy Invariance Theorem

Suppose for some domain , and are two piecewise paths in . Let be a homotopy from to . Let and be the paths followed by the initial and final points of the homotopy respectively. Then

Proof The required equation is equivalent to This is equivalent to say that showing that the integral around the outside of the “rectangle” is zero. We will chop it into pieces of small rectangles. We write for the inner small “rectangle” whose bottom left corner is , and write for . Thus we have So it suffices to argue that each of these integrals is zero. We now formalize the above setting, define a family of paths by Furthermore, let And . Since is the image of a compact set under a continuous map, is also compact (see theorem). Applying the previous lemma, there exists , such that for all , . Since is compact, by theorem, is uniformly continuous, so there is such that when and we have . Now we choose such that . Then all satisfy Therefore , hence . So by local existence of primitives, the closed path integral , and the result follows.

Corollary

Suppose for some domain . If and are CP-homotopic or PCE-homotopic, then

Proof If and are CP-homotopic, then , and we get the required result. If they are PCE-homotopic, then and are constant, implying .

e.g. and are not CP-homotopic in . We can justify this by observing that while .

Simply Connected Domains

Simply Connected Domain

A domain is called simply connected if every closed path in is CP-homotopic to a constant path.

Simply Connected Domain Characterisation

For a domain , the following are equivalent:

• is simply connected.
• is connected.
• in is connected.

Moreover, if is bounded, then these are also equivalent to

• is connected.
• is connected.

e.g. is not simply connected since is not connected.

Proposition

If is simply connected and , then for every closed we have

Proof Since on a simply connected domain, every is CP-homotopic to a constant path , then .

Corollary

If is simply connected and , then has a primitive.

Proof Directly from the theorem.

Proposition

Continuous maps preserves simple connectedness.