What is K-Theory?

Preface

Last article was about Combinatorics and the way it connects to the Champions League, this time this article is about K-Theory. (Yep, very out of the blue)

Level of the article:

Postgraduate Mathematics.

What is K-Theory?

K-theory is one of the key techniques both in Mathematics and Theoretical Physics, mainly used to classify objects according to certain equivalence relations, usually connected with vector bundles, modules, or algebraic varieties.

K-Theory in Mathematics:

Topological K-Theory:

Classification of Vector Bundles: Topological K-theory classifies vector bundles over a topological space. So, given a topological space, the idea is to understand all possible vector bundles over it, at least up to isomorphism.

Algebraic K-Theory:

Projective modules classification: Algebraic K-theory generalizes the notion of classifying vector bundles to that of classifying projective modules over a ring, modules resembling vector spaces but in a more general algebraic setting.

Invariants of rings: It establishes and makes available a vast wealth of knowledge of invariants of the rings and the schemes — cardinal in modern algebraic geometry.

In Theoretical physics:

String Theory: K-theory is used to classify D-branes in string theory — objects where open strings can end. D-branes are crucial in understanding the non-perturbative features of string theory.

History of K-Theory

The history of K-theory is rich and involves significant developments in both pure mathematics and its applications to physics.

Early Beginnings (1950s — 1960s):

Grothendieck’s Contribution (1957): The origins of K-theory are often traced back to Alexander Grothendieck. He introduced the concept in the late 1950s in the context of algebraic geometry. Grothendieck defined what is now known as Grothendieck groups to formalize the notion of vector bundles over algebraic varieties. His work was motivated by the need to classify vector bundles up to isomorphism, and he introduced what we now call algebraic K-theory to study these classifications systematically.

Atiyah and Hirzebruch (1959–1961): Around the same time, Michael Atiyah and Friedrich Hirzebruch developed topological K-theory. They were motivated by the need to study vector bundles in a topological rather than algebraic context. Their work extended the ideas of Grothendieck to topological spaces and involved applying homotopy theory to classify vector bundles. Their collaboration led to the Atiyah-Hirzebruch spectral sequence, which connects topological K-theory with ordinary cohomology theories.

Foundational Developments (1960s — 1970s)

Bott Periodicity Theorem (1959–1966): A key breakthrough was the Bott periodicity theorem, discovered by Raoul Bott in 1959. This theorem showed that the K-theory of spheres exhibits an 8-fold periodicity, which greatly simplified the computation of K-theory groups.

Algebraic K-Theory and Milnor (1970)

John Milnor contributed to algebraic K-theory by developing a new framework for understanding higher K-theory groups, which go beyond the original Grothendieck group. His work laid the groundwork for understanding more complex algebraic structures. This led to the definition of higher algebraic K-groups Kₙ (R), for a ring R, which are now central objects of study in algebraic K-theory.

Expansion and Applications (1970s — 1980s):

Applications in Index Theory: In the 1970s, K-theory found significant applications in index theory. Atiyah and Singer’s index theorem, which relates the analytical index of an elliptic differential operator to its topological index, was a major milestone that deeply connected K-theory with analysis and differential geometry.

C*-Algebras and Noncommutative Geometry: The 1970s and 1980s saw K-theory being applied to C*-algebras, initiated by George Mackey and Gennadi Kasparov. This branch, often referred to as operator K-theory, played a crucial role in the development of noncommutative geometry. Alain Connes later used K-theory as a fundamental tool in noncommutative geometry, where it helps to classify noncommutative spaces analogous to the classification of topological spaces.

Modern Developments and Physics (1990s — Present):

K-Theory in String Theory: In the 1990s, K-theory made its way into theoretical physics, particularly in string theory. Edward Witten and others showed that K-theory could be used to classify D-branes, which are crucial objects in string theory where strings can end.

First Steps in K-Theory

Before doing any type of K-theory, we need first to study the mathematical objects that this theory studies: Vector Bundles. In this article, we will not be able to define the main classical notion of K-Theory, which is the Grothendieck group. However, we will still learn some useful mathematics that can be used in fields such as Differential Geometry and Algebraic Topology.

Let k be the field of the real number or the complex numbers, then we have the following definition of a quasi-vector field.

Definition (Quasi-Vector Field)

A quasi-vector bundle with base X is given by:

1) A finite dimensional k-vector space Eₓ for every point x of X.

2) A topology on the disjoint union E= ∐ₓ Eₓ which induces the natural topology on each Eₓ, such that the natural projection 𝜋: E ⟶ X is continuous.

Notation

A quasi-vector bundle is denoted by 𝜉 = (E, 𝜋, X). The space E is the total space of 𝜉 and Eₓ is the fiber of 𝜉 at the point x.

Example 1

I like to think about of quasi-vector bundles as all the tangent vector to Sᶰ provides a nice geometric study, to put it more formally let X be the sphere Sᶰ= {x ∈ ℝⁿ⁺¹ : |x|=1}. For every point x of Sᶰ we choose Eₓ to be the vector space orthogonal to x, then E= ∐ₓ Eₓ, is the space of all the tangent vectors in Sᶰ.

After defining the spaces 𝜉 the next step is to define the morphisms between them, for these we have the following definition:

General Morphism:

Let 𝜉 = (E, 𝜋, X) and 𝜉’ = (E’, 𝜋’, X’) be quasi-vector bundles. A general morphism from 𝜉 to 𝜉 ‘ is given by a pair (f, g) of continuous maps f: X ⟶X’ and g: E ⟶E’ such that:

1) The diagram is commutative

2) The map gₓ: Eₓ ⟶ E’_f₍ₓ₎ induced by g is k-linear.

We now give a definition of a concept which will enable us to define what a vector bundle is, the concept of the Trivial Vector Bundles.

Definition of Trivial Vector Bundles

Let V be a finite dimensional vector space over k, if we have that E = X × V, and we have that for all x ∈ X we have Eₓ = V, then we call 𝜉 = (E, 𝜋, X) a Trivial vector bundle.

Example

Let TSᶰ denote the quasi-vector bundle considered in example 1. It is a very deep result that TSᶰ is not isomorphic to a trivial bundle unless n=1,3, or 7. (Check the Sources)

For results involving quasi-vectors, check the reference in section 1; Now that we know a bit about quasi-vector bundles, we will introduce vector bundles.

Vector Bundles

Definition.

Let 𝜉 = (E, 𝜋, X) be a quasi-vector bundle. Then 𝜉 is said to be a vector bundle if, for every point x in X, there exists a neighbourhood U of x such that 𝜉ᵤ is isomorphic to a trivial bundle.

Remark

This condition can be expressed in a more explicit way as if there exists a finite dimensional vector space V and a homeomorphism 𝜑: U ×V ⟶ 𝜋 ⁻¹(U) such that the diagram commutes

and such that for every point y ∈ U the map 𝜑_y: V ⟶ E_y is k-linear.

We call U a trivialization domain of the vector bundle 𝜉 and a cover (Uᵢ) of X is called a trivialization cover if each Uᵢ is a trivialization domain.

Future Content on K-Theory

For future content, we will develop the notion of Vector Bundle and explore some properties of it with the objective of reaching the Grothendieck group.

Challenges

Challenge 1

Prove that the quasi-vector bundle given in Example1 in Sᶰ is a vector bundle.

Solution:

See section 1.2 of the sources.

Challenge 2

Think of a quasi-vector bundle that is not a vector bundle.

Solution:

See section 1.2 of the sources

Sources

M. Karoubi, K-Theory: An Introduction, 1st ed., vol. 226, Classics in Mathematics. Berlin, Heidelberg: Springer-Verlag, 1978, pp. XVIII, 316. doi: 10.1007/978–3–540–79890–3.

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