Homotopy Types

Homotopy

Retraction

Suppose is a topological space, then a continuous map such that is called a retraction. The image of is called the retract of .

Remark

Retraction is the topological analogue of a projection in linear algebra. In general, they are the maps that “project” the space onto a subspace while preserving the structure of the space.

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 .

e.g. A deformation retraction of onto a subspace is a homotopy from the identity map of to a retraction of onto , that is, a map such that and .

Relative Homotopy

Let be a subspace. be continuous maps with . Then we say relative to if there exists a homotopy such that for all and .

Homotopy Equivalence

A map is called a homotopy equivalence if there is a map such that and . In this case, topological spaces and are said to be homotopy equivalent, or to have the same homotopy type, and denoted .

Remark

It is easy to see that homotopy equivalence is an equivalence relation. Furthermore, it is weaker than homeomorphism.

Lemma

Two topological spaces are homotopy equivalent if and only if there exists a third space containing both and as deformation retracts.

Contractible Space

A topological space is contractible if it has the same homotopy type as a point.

Path Homotopy

An important concept in homotopy theory is the path homotopy, which deals with paths in topological spaces. We first recall the following definition of a path:

Let be a topological space. A path in joining two points is a continuous function such that and . If the start point and end point coincide, we call it a loop.

Link to original

So a path homotopy is a homotopy between paths relative to the boundary of the interval. Formally, given two paths in a topological space with same endpoints, a homotopy between them is a continuous function such that and for all .

Concatenation of Paths

For two paths and such that the end point of coincides with the start point of , we define their concatenation by

Reversed Path

If is a path, then the reversed path is given by .

Now we have some operations on paths, so one might want to ask if there is any algebraic structure on the set of paths. The answer is yes, and it is called the fundamental group, which will be discussed in the next page.