Mapping spaces

For algebraic objects, the most basic structure-preserving map was a homomorphism. The most basic equivalence in topology is the similarly named homeomorphism. Homeomorphic spaces are “the same” from a topological point of view.

 ◊ One can visualize a homeomorphism as stretching and bending a space arbitrarily, since length and curvature involve structure beyond open sets and so are “invisible” from the topological viewpoint.

There exists an even looser equivalency in topology, called a homotopy equivalency. The essential conceptual difference is that since it is bijective, a homeomorphism preserves the dimensionality of the object while stretching and bending, while a homotopy equivalency allows the collapse of dimensions, while still preserving holes.

To make this concept precise, we first define a homotopy between spaces, a continuous family of continuous maps $${f_{t}\colon X\to Y;}$$ i.e. $${f}$$ is continuous when considered as a function of $${t}$$ as well as when considered as a function of points in $${X}$$. Two maps are homotopic if there is a homotopy between them, i.e. $${f_{0}}$$ is homotopic to $${f_{1}}$$ if $${t}$$ runs from 0 to 1.

The above depicts a homotopy $${f_{t}}$$ from $${X}$$ to $${Y}$$; the map $${f_{0}}$$ onto $${Y}$$ is homotopic to the map $${f_{1}}$$ which maps all of $${X}$$ to a point $${y}$$.

A couple of related definitions are:

• Homotopy relative to $${A}$$ (denoted homotopy rel $${A}$$): a homotopy that is independent of $${t}$$ on $${A\subset X}$$; e.g. in the above figure $${f_{t}}$$ is a homotopy rel $${x}$$
• Deformation retraction from $${X}$$ to $${A}$$: a homotopy rel $${A}$$ from the identity to a retraction from $${X}$$ to $${A}$$; e.g. if we take $${Y=X}$$ and $${y=x}$$ above, $${f_{t}}$$ is a deformation retraction from $${X}$$ to $${x}$$

The precise definition of two spaces being homotopy equivalent, or having the same homotopy type (denoted $${X\simeq Y}$$), is not important for our purposes; instead, we will state the derived fact that $${X\simeq Y}$$ iff they are deformation retracts of the same space. A space homotopy equivalent to a point is then called contractible.

The above depicts a deformation retraction that collapses the equatorial disc inside the sphere, the strings, and the solid cube to points; the two spaces are thus homotopy equivalent.

 ◊ Homotopy equivalency can be viewed as meaning that we can collapse or expand any parts of the space as well as bending and stretching.