Bordismusgruppen

Bordismusgruppen, also known as bordism groups, are a fundamental concept in differential topology. They are built from orientable manifolds and their cobordism classes. Two orientable n-dimensional manifolds, M and N, are said to be cobordant if there exists an orientable (n+1)-dimensional manifold W with boundary $\partial W = M \cup N$, where the union is understood as a disjoint union. This relation is an equivalence relation, and the equivalence classes are called bordism classes.

The set of all bordism classes of n-dimensional orientable manifolds forms an abelian group under the operation of disjoint union. This group is denoted by $\Omega_n(X)$ if the manifolds are further restricted to be embedded in a topological space X, or more generally, as $\Omega_n$ for closed orientable manifolds. The identity element of this group is the bordism class of the empty manifold, and the inverse of a bordism class $[M]$ is the bordism class of $M$ with the opposite orientation, $[-M]$.

The bordism groups play a crucial role in understanding the topology of manifolds. For instance, the first bordism group $\Omega_1$ is trivial, meaning any closed orientable 1-manifold is cobordant to the empty set. However, higher bordism groups can be quite rich. The calculation of these groups often involves techniques from algebraic topology, such as spectral sequences. For example, the complex bordism groups, denoted by $MU_*(X)$, are closely related to stable homotopy groups of spheres via the Conner-Floyd isomorphism. The study of bordism groups has led to significant advancements in the classification of manifolds and has connections to other areas of mathematics like algebraic geometry and mathematical physics.