Kobordismustheorie

Kobordismustheorie is a fundamental concept in differential topology and algebraic topology. It is a way of classifying manifolds up to cobordism. Two manifolds are considered cobordant if they are the boundary of a higher-dimensional manifold. This concept is particularly useful for understanding the structure and properties of manifolds.

The cobordism group of n-dimensional manifolds, denoted by $\Omega_n$, is the set of all oriented closed n-dimensional manifolds with the equivalence relation of cobordism. The operation in this group is the disjoint union of manifolds. The cobordism group is an abelian group. For example, the cobordism group of 1-dimensional manifolds, $\Omega_1$, is trivial, meaning all closed 1-manifolds (which are just circles) are cobordant to the empty set. The cobordism group of 2-dimensional manifolds, $\Omega_2$, is also trivial. However, $\Omega_3$ is not trivial and is isomorphic to $\mathbb{Z}_2$.

The cobordism theory has deep connections to characteristic classes and homology theory. The Thom isomorphism theorem is a crucial result that relates cobordism groups to the homotopy groups of Thom spectra. This connection allows for the computation of cobordism groups using algebraic topology tools. The study of cobordism also leads to the development of generalized homology and cohomology theories, known as cobordism theories. These theories can be used to define topological invariants and study the structure of spaces that may not be manifolds. It plays a significant role in understanding the classification of topological spaces and has applications in areas such as string theory and general relativity.