---
title: K-Theory
description: K-theory is an algebraic framework used to study vector bundles on topological spaces or projective modules over rings. Learners will understand how to compute K-groups to solve problems in topology, geometry, and operator algebras.
category: mathematics
subcategory: algebraic-topology
difficulty: beginner, intermediate, advanced
url: /subject/k-theory
---

# K-Theory

K-theory is an algebraic framework used to study vector bundles on topological spaces or projective modules over rings. Learners will understand how to compute K-groups to solve problems in topology, geometry, and operator algebras.

## Available Resources

1 Books • 4 Websites

## Websites

### 1. Hatcher's Vector Bundles and K-theory

Allen Hatcher's free book-length notes on vector bundles, characteristic classes, and topological K-theory, including Bott periodicity and the Adams operations. Readers with algebraic topology background learn to compute K-groups and apply them to classical problems like Hopf invariant one.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf

**Tags:** topological-k-theory, vector-bundles, bott-periodicity, characteristic-classes, algebraic-topology

### 2. nLab: K-theory

The nLab wiki entry on K-theory, a collaborative research-level reference surveying topological, algebraic, and operator K-theory and their formulation as generalized cohomology theories and spectra. Useful for seeing how the variants relate within higher category theory.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://ncatlab.org/nlab/show/K-theory

**Tags:** k-theory, algebraic-k-theory, generalized-cohomology, spectra, higher-category-theory

### 3. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

### 4. nLab

Collaborative wiki for mathematics, physics and philosophy written from a category-theoretic viewpoint, with cross-linked entries on categories, functors, adjunctions, topos theory, higher category theory and homotopy theory. Readers can look up precise definitions, examples and references for advanced structural mathematics.

**Difficulty:** Advanced | **Language:** English | **Price:** Free

**Link:** https://ncatlab.org

**Tags:** category-theory, higher-category-theory, topos-theory, homotopy-theory, reference-wiki

## Books

### 1. K-Theory

**Author:** Michael Atiyah

Atiyah's classic lecture notes on topological K-theory, built from vector bundles over compact spaces. Covers the Bott periodicity theorem, operations on K-groups and Adams operations, preparing readers to use K-theory as a generalized cohomology theory in algebraic topology.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0201093944?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, algebraic-topology

---

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