Homotopy Equivalence - Homotopy Type

We have now introduced an equivalence relation on the class of all continuous functions, which allows us to think about two maps as the same if they can be continuously deformed into each other. This enables us to define a similar concept for topological spaces.

Definition 5.10   Let X and Y be spaces. We say that X and Y are homotopy equivalent if there are maps f : X $\rightarrow$ Y and g : Y $\rightarrow$ X such that fg $\simeq$ id_Y and gf $\simeq$ id_X.

Intuitively, X and Y are homotopy equivalent if each can be deformed into the other in a continuous manner. The first observation one makes is the following.

Lemma 5.11   Homotopy equivalence is an equivalence relation on spaces.

Homotopy equivalence of spaces also has a relative version.

Definition 5.12   Two pairs (X, A) and (Y, B) are said to be homotopy equivalent if there are maps of pairs f : (X, A) $\rightarrow$ (Y, B) and g : (Y, B) $\rightarrow$ (X, A) such that fg $\simeq$ id_Y relA and gf $\simeq$ id_X relB.

We proceed with some easy examples.

Example 5.13   Every convex subset of Rⁿ is homotopy equivalent to a single point. To see this let B be a convex subset of Rⁿ. Let b₀ $\in$ B be any point. For an arbitrary point b $\in$ B the line segment (1 - t)b + tb₀, 0 $\leq$ t $\leq$ 1 is contained in B. Let pt denote the one point space. Let f : pt $\rightarrow$ B denote the map taking the single point in pt to b₀ and let g : B $\rightarrow$ pt denote the constant map. Then gf is the identity on pt (and thus obviously homotopic to the identity). Define a homotopy

H : B x I $$\rightarrow$$ B

by H(b, t) = (1 - t)b + tb₀. Then H is continuous (convince yourself) and satisfies H(b, 0) = b and H(b, 1) = b₀ and so H is a homotopy of fg to the identity map of B.

A space which is homotopy equivalent to a single point space is called contractible. There are more contractible spaces than just convex subsets of Rⁿ. Even among subsets of Euclidean space there are many subspaces which are contractible without being convex, as one can imagine. Intuitively a subset of Rⁿ is contractible if and only if it has no holes. Writing an explicit homotopy equivalence however may be rather difficult.

Example 5.14   The solid torus and the cylinder are both homotopy equivalent to the circle S¹. To see this notice that the solid torus is just the Cartesian product S¹ x D² and D² is contractible. The Cylinder is S¹ x I and I is contractible. Thus both assertions follow from the lemma below.

Lemma 5.15   Let X be any space and let C be a contractible space. Then X x C is homotopy equivalent to X.

The relation of homotopy equivalence divides spaces into equivalence classes. For a given space X the equivalence class of X under homotopy equivalence is called the homotopy type of X. Thus two spaces are said to have the same homotopy type if they are homotopy equivalent. Homotopy type is a much weaker topological invariant than topological type, the class of a space under topological equivalence. However it is an extremely useful generalisation. Namely, two spaces that are not of the same homotopy type can never have the same topological type. Thus for example, to show that all the surfaces constructed in the classification theorem are indeed distinct, it suffices to show that they are not of the same homotopy type.

The homotopy type of a space is easier to determine than its homeomorphism type. However even this weaker invariant is an extremely rich one and in general the task of distinguishing two homotopy types is hopeless. The study of homotopy types is the main theme of ``homotopy theory''. In the next sections we will familiarise ourselves with some of the main tools in the subject.