Bordismustheorien

Bordismustheorien, often translated as bordism theory, is a significant branch of algebraic topology that studies the structure of manifolds in relation to their boundaries. At its core, bordism theory classifies manifolds by considering when one manifold can be "filled in" by a higher-dimensional manifold whose boundary is the original manifold.

Specifically, two closed manifolds M and N are said to be bordant if there exists a compact manifold W with boundary ∂W = M ∪ N (where the union is disjoint and the orientation of M is reversed) such that M and N are homologous as submanifolds of W. This relation is an equivalence relation, and the equivalence classes are called bordism classes. The set of all bordism classes of a certain dimension, equipped with a natural addition operation (disjoint union), forms a group called the bordism group.

Bordism theory provides powerful tools for understanding topological invariants of manifolds. For instance, the cobordism group of oriented n-dimensional manifolds, denoted by Ω_n(X) for a topological space X, captures information about the characteristic classes of vector bundles over X. Thom's theorem established a profound connection between bordism groups and stable homotopy groups of spheres, showing that Ω_n is isomorphic to the stable homotopy group π_n(MO), where MO is the Thom spectrum of oriented manifolds. This connection is fundamental to modern algebraic topology and has numerous applications in geometry and physics.