Algebraic v. topological K-theory: a friendly match

These notes evolved from the lecture notes of a minicourse given in Swisk, the Sedano Winter School on K-theory held in Sedano, Spain, during the week January 22–27 of 2007, and from those of a longer course given in the University of Buenos Aires, during the second half of 2006. They intend to be an introduction to $K$-theory, with emphasis in the comparison between its algebraic and topological variants. We have tried to keep as elementary as possible.

💡 Research Summary

The notes, compiled from a winter school in Sedano (January 2007) and a semester‑long course in Buenos Aires (2006), serve as an introductory yet systematic comparison of algebraic and topological K‑theory. After a brief motivation, the authors present topological K‑theory for compact Hausdorff spaces X: K⁰(X) is defined as the Grothendieck group of isomorphism classes of complex vector bundles, and K¹(X) is introduced via suspension, K¹(X)=K̃⁰(ΣX). Bott periodicity is sketched, yielding the 2‑periodic structure Kⁿ(X)≅Kⁿ⁺²(X). The Atiyah‑Hirzebruch spectral sequence is explained as the bridge between ordinary cohomology and K‑theory, with concrete calculations for spheres, projective spaces, and CW‑complexes illustrating its utility.

The algebraic side begins with K₀(R) as the Grothendieck group of finitely generated projective R‑modules and K₁(R) as the abelianization of GL(R) (or π₁ of the classifying space). Higher groups Kₙ(R), n≥2, are introduced via Quillen’s Q‑construction and the plus‑construction, together with Bass’s stabilization results. The authors emphasize the parallelism: both theories start from a notion of “projective object” and then pass to a group completion, yet the underlying categories differ (topological bundles vs. algebraic modules).

A central theme is the comparison map. For a complex algebraic variety V, the analytification functor yields a natural homomorphism K₀^{alg}(V)→K⁰(V^{an}), which becomes an isomorphism after tensoring with ℚ via the Chern character. The notes also discuss the role of the Chern character in both settings, the relationship between K‑theory and cyclic homology, and brief remarks on modern extensions such as motivic K‑theory and K‑theory of non‑commutative C*‑algebras.

Pedagogically, each chapter ends with exercises ranging from elementary computations to proofs of key theorems, and a curated bibliography points readers toward classic texts (Atiyah‑Segal, Quillen, Weibel) and recent surveys. The final section outlines open problems and suggests a roadmap for advancing from the introductory material to current research topics, making the notes a useful bridge for graduate students moving between algebraic and topological perspectives on K‑theory.