18 Lectures on K-Theory

We present 18 Introductory Lectures on K-Theory covering its basic three branches, namely topological, analytic (K-Homology) and Higher Algebraic K-Theory, 6 lectures on each branch. The skeleton of these notes was provided by the author’s personal notes from a graduate summer school on K-Theory organised by the London Mathematical Society (LMS) back in 1995 in Lancaster, UK.

💡 Research Summary

The manuscript “18 Lectures on K‑Theory” is a comprehensive set of graduate‑level lecture notes that systematically introduces the three principal branches of K‑theory: topological K‑theory, analytic K‑homology, and higher algebraic K‑theory. Originating from the author’s personal notes used in a 1995 London Mathematical Society summer school, the material is organized into eighteen one‑hour lectures, six for each branch, providing a balanced blend of foundational definitions, central theorems, illustrative examples, and problem sets.

The first six lectures cover topological K‑theory. They begin with vector bundles, the construction of K⁰ and K¹ groups, and the fundamental properties of these groups. Bott periodicity is proved and its consequences for the computation of K‑groups of spheres, tori, and complex projective spaces are worked out in detail. The Atiyah–Hirzebruch spectral sequence and the relationship between K‑theory and characteristic classes via the Chern character are presented, giving students tools to extract homotopy‑theoretic invariants from geometric data.

Lectures seven through twelve turn to analytic K‑homology, the dual theory to topological K‑theory. After a concise review of Fredholm operators and the index theorem, the notes define K‑homology classes using cycles of the form (H, π, F) where H is a Hilbert space, π a representation of a C∗‑algebra, and F a Fredholm operator. The duality between K‑theory of C∗‑algebras and K‑homology is emphasized, and the construction of external products and the Poincaré duality map are explained. A concrete Dirac‑operator example on a spin manifold illustrates how geometric data give rise to K‑homology classes. The section also sketches the early form of the Baum–Connes conjecture, highlighting its role as a bridge between non‑commutative geometry and topology.

The final six lectures address higher algebraic K‑theory. Starting with Milnor’s K₁ and K₂, the notes quickly move to Quillen’s plus‑construction and Q‑construction, establishing the definition of Kₙ(R) for any ring R and any integer n ≥ 0. The relationship between Milnor K‑theory and Quillen K‑theory is clarified, and the Steinberg group is introduced to describe K₂. Computational techniques such as the use of spectral sequences, devissage, and localization sequences are demonstrated on examples like finite fields, Dedekind domains, and polynomial rings. The Chern character from algebraic K‑theory to cyclic homology is derived, and its role in connecting K‑theory to motivic cohomology is discussed. The last lecture surveys contemporary research directions, including Vaserstein’s conjecture, the interaction between higher K‑theory and higher K‑homology, and open problems related to the Farrell–Jones conjecture.

Each lecture follows a consistent structure: a concise statement of definitions, a “key theorem” box with proof sketches, a set of worked examples, and a collection of exercises with hints. The author supplements the text with numerous diagrams, tables of K‑groups for standard spaces and rings, and a curated bibliography that points to classic sources (Atiyah–Singer, Blackadar, Weibel) as well as recent survey articles. This pedagogical design makes the notes suitable both as a self‑study guide for graduate students entering K‑theory and as a quick reference for researchers needing a refresher on the basic machinery across the three major flavors of the subject.