In differential topology, the transversality theorem, also known as the Thom transversality theorem after French mathematician René Thom, is a major result that describes the transverse intersection properties of a smooth family of smooth maps. It says that transversality is a generic property: any smooth map , may be deformed by an arbitrary small amount into a map that is transverse to a given submanifold . Together with the Pontryagin–Thom construction, it is the technical heart of cobordism theory, and the starting point for surgery theory. The finite-dimensional version of the transversality theorem is also a very useful tool for establishing the genericity of a property which is dependent on a finite number of real parameters and which is expressible using a system of nonlinear equations. This can be extended to an infinite-dimensional parametrization using the infinite-dimensional version of the transversality theorem. Finite-dimensional version Previous definitions Let be a smooth map between smooth manifolds, and let be a submanifold of . We say that is transverse to , denoted as , if and only if for every we have that . An important result about transversality states that if a smooth map is transverse to , then is a regular submanifold of . If is a manifold with boundary, then we can define the restriction of the map to the boundary, as . The map is smooth, and it allows us to state an extension of the previous result: if both and , then is a regular submanifold of with boundary, and . Parametric transversality theorem Consider the map and define . This generates a family of mappings . We require that the family vary smoothly by assuming to be a (smooth) manifold and to be smooth. The statement of the parametric transversality theorem is: Suppose that is a smooth map of manifolds, where only has boundary, and let be any submanifold of without boundary. If both and are transverse to , then for almost every , both and are transverse to . More general transversality theorems The parametric transversality theorem above is sufficient for many elementary applications (see the book by Guillemin and Pollack). There are more powerful statements (collectively known as transversality theorems) that imply the parametric transversality theorem and are needed for more advanced applications. Informally, the "transversality theorem" states that the set of mappings that are transverse to a given submanifold is a dense open (or,