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.


An Illustrated Handbook