Automorphisms of Manifolds and Algebraic K-Theory: Part III

TL;DR

This paper develops a highly connected map linking the structure space of a closed manifold to algebraic K- and L-theory, refining surgery theory analysis of manifold structures.

Contribution

It introduces a new construction that connects the structure space of manifolds with algebraic K- and L-theory, enhancing understanding of manifold automorphisms.

Findings

Established a highly connected map from structure space to algebraic K- and L-theory spaces.

Refined the surgery theoretic analysis of block structure spaces.

Provided new tools for studying automorphisms of manifolds.

Abstract

The structure space S(M) of a closed topological m-manifold M classifies bundles whose fibers are closed m-manifolds equipped with a homotopy equivalence to M. We construct a highly connected map from S(M) to a concoction of algebraic L-theory and algebraic K-theory spaces associated with M. The construction refines the well-known surgery theoretic analysis of the block structure space of M in terms of L-theory.