Lgroups

Lgroups, often written as L-groups, is a term used in mathematics to denote several families of invariants that arise in different areas of the subject. The most common usages appear in algebraic topology (surgery theory) and in the representation-theoretic Langlands program.

In algebraic topology and surgery theory, L-groups are abelian groups associated to a ring with involution. They are typically denoted L_n(R) for integers n and defined using quadratic or symmetric forms on chain complexes with Poincaré duality. These groups measure obstructions to modifying a space by surgical procedures so that it becomes homeomorphic to a target manifold within a given homotopy type. L-groups appear in the formulation of the surgery exact sequence, linking geometric questions about manifolds to purely algebraic data. Depending on the formulation, these groups exhibit periodicity (traditionally 8-periodic in many setups) and play a central role in classifying high-dimensional manifolds up to surgery.

In the Langlands program, the L-group, denoted ^L G, is a different construction associated with a reductive algebraic group G over a field. It is built from the dual group G^ and the Galois or Weil group of the base field, and is used as the target for Langlands parameters. The L-group encodes both representation-theoretic information of G and arithmetic information from the field, and it underpins the formulation of local and global Langlands correspondences, as well as related concepts such as L-packets and endoscopy.

The term L-group thus covers distinct but influential concepts in topology and number theory, united by a common naming convention rather than a shared definition.