Homotopy

Definition:

Given topological spaces and and continuous maps and , we say that is homotopic to if there exists a continuous map such that for all , and

If is homotopic to ,we use the notation

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Examples:

Example 1 Let be the natural inclusion map, and let be the constant map . Let

be defined by .

Example 2 Let be the circle defined by the parametric equation ,

and let be the ellipse .

Let

be defined by

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Theorem:

Homotopy is an equivalence relation.

Proof:

1. : By the homotopy

2. : If is a homotopy from to then is a homotopy from to

3. and

Given a homotopy from to and a homotopy from to we define a function

for

and

for

Note that this is well defined since

What is left to show is that the function is continuous. To do this we invoke the push out construction.

Let and be two topological space homeomorphic to but disjoint from and each other. Consider the push out diagram

defined by for

and for

We are implicitly using the various homeomorphisms between , , and

Consider the diagram:

We have at once that is continuous since

for

and

for

What still needs to be verified is that the identity map is a homeomorphism between with the push out topology and with the product topology.

To see this we look at the diagram

where the right most is considered to have the product topology. Again,

Homework Due Thursday April 27

verify that is continuous hence is continuous.

and,

under ,the image of closed sets are closed hence the inverse of is continuous with respect to the two topologies.