MSGSS
Algebraic L-Theory via Poincare Infinity Categories
Algebraic L-theory was developed in the late 1960’s by C.T.C. Wall and others as a means of classifying higher dimensional manifolds via surgery theory. Importantly, given a triangulated topological space with Poincare duality of dimension greater than four, the theory defines a sequence of L-groups that encode the obstructions to surgery, which in turn tells us if the space is homotopic to a manifold. The theory was refined throughout the 70’s and 80’s by A. Ranicki, who systematically formulated L-theory in terms of chain complexes, depending only on a ring with involution, and in so doing managed to prove a Morita equivalence theorem for L-groups and also that the L-groups form a spectrum with four-periodicity.
However, despite Ranicki’s successes, his chain complex formulation of the L-groups can be difficult to use in practice, and the proofs of the aforementioned theorems are somewhat challenging. Luckily, more recent joint work by Baptiste Calmès, Markus Land, Thomas Nikolaus, and six others has once again reformulated algebraic L-theory using the notion of a Poincare infinity category. The authors show that this reformulation does indeed recover all of Ranicki’s theory, and furthermore it shows that one can do L-theory on any Poincare infinity category, providing a substantial generalization of the theory.
In this talk, I will introduce Ranicki’s chain complex formulation of L-theory and outline the challenges associated with it. I will then introduce the infinity categorical formulation of Calmès et al and demonstrate that this approach is not only more general than the original theory, but also more natural. Time permitting, I will also briefly outline possible avenues of future research inspired by this modern categorical formulation of L-theory.