Homotopy and Deformation: The Shape of Continuity

March 17, 2026 · Luciano Muratore

Homotopy and Deformation

In topology, we often ask: When are two shapes essentially the same? While general topology focuses on “one-to-one” stretching (homeomorphisms), algebraic topology introduces a more flexible perspective. It allows us to “squash” and “contract” spaces, focusing on the features that remain invariant under continuous motion.

The Concept of Homotopy

The fundamental tool for describing motion in topology is Homotopy.

Formally, a homotopy is a continuous family of maps $f_t: X \to Y$ indexed by the interval $t \in [0,1]$ . This is represented by a single joint continuous function: $F: X \times [0,1] \to Y$ where $(x,t) \mapsto f_t(x)$ .

Intuition: Think of a homotopy as a “movie” where each frame is a continuous function. By indexing on $t$ , we can “see” or keep track of the deformation as it evolves through time, transforming an initial function $f_0$ into a final function $f_1$ .

Homotopy Relative to a Subset

Sometimes, during a deformation, we want certain parts of the space to remain “pinned down.”

If a homotopy satisfies that $f_t|_A$ is independent of $t$ for some subset $A \subset X$ , we call it a Homotopy relative to A.

Intuition: There is a specific part of the space $X$ that remains immutable during the deformation. While the rest of the map “wiggles” or shifts, the points in $A$ act as fixed anchors.

Retractions and Deformation Retractions

To understand how a large space can be represented by a smaller one, we use Retractions.

A Retraction is a continuous map $r: X \to X$ such that $r^2 = r$ . This algebraic property signifies that the map acts as a projection: once a point is moved into the image, it stays there.

A Deformation Retraction is the “movie version” of this projection. It is a homotopy between the identity map and a retraction, where the subspace being retracted onto remains fixed throughout the process.

Intuition: A retraction is the final “squash,” while the deformation retraction represents the actual motion of the points of the space as they flow toward the subspace. The projected space changes, but the target “skeleton” stays the same.

The Mapping Cylinder: Factoring Any Map

One of the most powerful results in this area is that every continuous map $f: X \to Y$ can be “tamed” by viewing it through a geometric structure called a Mapping Cylinder ( $M_f$ ).

We can factor any map $f$ as:

An Inclusion of $X$ into the cylinder.
A Retraction (via a deformation retraction) from the cylinder onto $Y$ .

Intuition: By “gluing” a cylinder between $X$ and $Y$ via the map $f$ , we turn a potentially messy function into a geometric flow. We include the initial space into a new structure and then retract that structure onto the target. This shows that every map is essentially a deformation.

Homotopy Equivalence and Contractibility

The most relaxed form of “sameness” in topology is Homotopy Equivalence. Two spaces $X$ and $Y$ are homotopy equivalent if they can be deformed into one another.

A special case of this is Contractibility. A space is contractible if it is homotopy equivalent to a single point.

Example: $\mathbb{R}^n$ vs. The Circle

$\mathbb{R}^n$ is contractible: We can define a straight-line homotopy where everyone in the space walks toward the origin simultaneously. At the end of the “movie,” the entire infinite space has become a single point $\{0\}$ .
The Circle ( $S^1$ ) is NOT contractible: You cannot shrink a circle to a point without “breaking” it. Breaking the circle implies a loss of continuity; there is no way to move all points to a single center without a “snap” because of the hole in the middle.

The Punctured Circle: Interestingly, if we remove just one point from a circle ( $S^1 \setminus \{x\}$ ), the “cycle” is broken. It becomes topologically equivalent to an interval, which is contractible to a point.

Summary

Homotopy is the continuous evolution of a function over time.
Relative Homotopy keeps a specific subset fixed during the change.
Retractions squash a space onto a subspace; Deformation Retractions show the movement of that squash.
Mapping Cylinders allow us to treat any continuous map as a geometric deformation.
Homotopy Equivalence ignores “squashing” and only cares about the fundamental structure (holes).
Contractibility describes spaces that can be shrunk to a single point without tearing.