Homotopy

Homotopy is a fundamental concept in algebraic topology used to study the continuous deformation of functions and spaces. In topology, we often want to know if one shape or mapping can be transformed into another without tearing or breaking. When two continuous functions can be smoothly changed from one into the other, they are called homotopic. This idea allows mathematicians to categorize shapes and mappings based on their essential properties rather than their exact measurements.

To understand the mechanism of a homotopy, imagine a slider control that moves from 0 to 1. Formally, a homotopy between two continuous functions, $f$ and $g$, is a continuous function $H$ that maps the product of a space $X$ and the unit interval $[0, 1]$ into a space $Y$. We can think of the second parameter, $t$, as time. At time $t=0$, the function is $f$. As the slider moves toward $t=1$, the function undergoes a continuous deformation. By the time the slider reaches 1, the function has become $g$.

Mathematicians use homotopy to define homotopy equivalence between two different topological spaces, $X$ and $Y$. Two spaces are homotopy equivalent if there exist continuous maps between them that can be deformed back into the identity maps of those spaces. This means the spaces share the same "homotopy type." You can think of this as transforming spaces through bending, shrinking, or expanding. A special case of this is a contractible space, which is a space that can be continuously shrunk down to a single point. For example, a solid disk is homotopy equivalent to a point because it can be deformed along radial lines.

It is important to distinguish homotopy equivalence from homeomorphism. A homeomorphism is a much stricter type of equivalence where the two maps between spaces must be exact inverses of each other. While all homeomorphic spaces are homotopy equivalent, the reverse is not always true. A Möbius strip and an untwisted strip are homotopy equivalent because both can be deformed into a circle. However, they are not homeomorphic. This distinction helps topologists understand which properties are deep and which are superficial.

Homotopy is also used to define more specific types of transformations, such as isotopy. An isotopy is a special kind of homotopy where each intermediate step remains an embedding. In simpler terms, the shape cannot pass through itself during the deformation. This is a critical concept in knot theory. When mathematicians ask if two knots are the same, they often look for an ambient isotopy. This involves moving the entire surrounding space to transform one knot into another without cutting the string.

Another variation is homotopy relative to a subspace, denoted as $K$. In this version, certain parts of the space are required to remain fixed during the entire deformation. This is used to define the fundamental group, which is a key tool in topology. If the subspace $K$ is just a single point, the process is called a pointed homotopy. These variations allow researchers to study how paths and loops behave within complex structures while keeping certain boundaries stable.

The significance of homotopy lies in its role as an invariant in algebraic topology. Many mathematical concepts are homotopy invariant, meaning they do not change if the space undergoes a homotopy equivalence. For instance, if two spaces are homotopy equivalent, they will share the same path-connectedness and simple connectedness. They also share isomorphic homology, cohomology, and fundamental groups. These groups act as mathematical fingerprints that describe the holes and connectivity of a space.

Beyond basic shapes, homotopy connects to advanced physics and geometry. On a Lorentzian manifold, mathematicians study timelike homotopy. This involves curves that represent paths moving forward in time. A manifold might be simply connected by standard curves but still be multiply connected by timelike curves. This demonstrates how the specific rules of a space, like the flow of time, change how we apply the concept of homotopy to understand the universe.