L-theory

L-theory is a collection of algebraic methods in topology and K-theory used to classify manifolds, particularly in relation to surgery theory. It provides an algebraic invariant that determines the obstruction to performing surgery on a map to make it a homotopy equivalence. More precisely, it studies the Witt group of quadratic forms on modules with chain complexes.

L-theory exists in several variations, broadly categorized into:

• Surgery Obstruction Groups (L-groups): These groups, denoted L_n(R), depend on a ring with involution R and an integer n. They encode the obstructions to performing surgery on a degree 1 map of manifolds to obtain a simple homotopy equivalence. The subscript 'n' represents the dimension of the manifold being considered. There are different variants depending on the type of equivalence desired (homotopy or simple homotopy), which affects the definition of the L-groups. These groups are fundamental to the surgery exact sequence, which relates the structure set of a manifold to its normal invariants and the surgery obstruction groups.

• Algebraic L-theory: This provides a more abstract algebraic formulation of the concepts in the surgery obstruction groups. Instead of focusing directly on manifolds and surgery, algebraic L-theory studies quadratic forms on modules and chain complexes more generally. This allows for applications in a broader range of contexts, including the study of Poincaré duality spaces.

• Topological L-theory: Relates to the topological category (TOP).

• Smooth L-theory: Relates to the smooth category (DIFF).

The connection to surgery theory arises because L-theory provides an algebraic invariant that captures the difference between the chain complex of a manifold and that of a simpler space (like a sphere). If this invariant vanishes, then surgery can be performed to transform the manifold into the simpler space.

The computation of L-groups is a central problem in surgery theory. While there are general methods, the specific calculations often depend heavily on the ring with involution R under consideration. Important cases include the integers (Z), group rings of groups (Z[π]), and fields.

L-theory is closely related to K-theory. Both are fundamental tools in algebraic topology and play a crucial role in the classification of manifolds. They also have connections to other areas of mathematics, such as number theory and representation theory.