Deformation in topology is a homotopy, which is a class of maps. In mathematics, many types of maps or functions exist beyond the homotopy classes of maps in topology. Here are a few:

1. Bijective Map: A map between two sets is bijective (or a bijection) if every element of each set is paired with exactly one element of the other set in such a way that every element of the other set is paired with exactly one element of the first set.
2. Embedding: An embedding is a map that preserves the properties of one space within another. In topology, for example, an embedding is a continuous map that is also a homeomorphism onto its image. Consider the real numbers R as a topological space, and the space R² of ordered pairs of real numbers. The function f(x) = (x, 0) is an embedding of R in R². It preserves all the properties of R, like continuity and closeness, within R².
3. Isomorphism: In many areas of mathematics, an isomorphism is a map that preserves both structure and operations, meaning that it’s a way to translate one mathematical structure into another of the same type, without losing information. For a simple example consider two vector spaces V and W, where V = {(a,b) | a, b in R} and W = {(c,d) | c, d in R}. The function f: V -> W defined by f((a,b)) = (a, a + b) is an isomorphism, it maintains all of the properties of the structures involved (addition and scalar multiplication).
4. Homeomorphism: This is a concept in topology. A homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. An example of a homeomorphim is a stretch of a rubber band. Imagine a rubber band, you can stretch it, squash it, twist it, but as long as you don’t tear it, it remains essentially a “circle”. A homeomorphism formalizes this idea of being able to “adjust” a space without tearing it.
5. Diffeomorphism: In the study of differentiable manifolds in differential geometry, a diffeomorphism is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth. An example is the function f: R -> R, defined by f(x) = x³. The function is differentiable and its inverse, the cube root function, is also differentiable.
6. Holomorphic/Analytic functions: These are functions that are complex-differentiable at every point in their domain. These are often studied in complex analysis. An example is the function f: C -> C defined by f(z) = z². This function is complex differentiable at every point in its (complex) domain.
7. Endomorphisms and Automorphisms: An endomorphism is a map from a mathematical object to itself, preserving all the structure. An automorphism is an endomorphism that also has a two-sided inverse, and so is a bijection from the object to itself. An example in the context of linear algebra is a square matrix M acting on a vector space V by multiplication. It maps each vector in V to another vector in the same vector space, thus it’s an endomorphism. If it’s an invertible matrix, then it’s an automorphism.