2.2 Definition and properties of K(X)

K(X) is the Grothendieck group of complex vector bundles over X. It transforms the monoid of vector bundles into a group, allowing formal differences between bundles. This construction is key to understanding K-theory's role in studying vector bundles.

K(X) has important properties like abelian group structure, rank homomorphism, and tensor product-induced ring structure. These properties make K-theory a powerful tool for analyzing topological spaces through their vector bundles, connecting algebra and topology.

K-Theory Group for Spaces

Definition of K(X)

• K(X) is the Grothendieck group of the monoid of isomorphism classes of complex vector bundles over X under the direct sum operation
• Elements of K(X) are formal differences [E] - [F], where E and F are complex vector bundles over X
  • [E] denotes the isomorphism class of E
• The group operation in K(X) is induced by the direct sum of vector bundles: [E] + [F] = [E ⊕ F]
• The zero element in K(X) is represented by the trivial vector bundle of any rank
• The inverse of [E] in K(X) is represented by the formal difference [F] - [E]
  • F is any vector bundle such that E ⊕ F is isomorphic to a trivial bundle

Properties of K(X)

• K(X) is an abelian group under the operation induced by the direct sum of vector bundles
• The rank of a vector bundle induces a group homomorphism from K(X) to the integers Z
  • This homomorphism sends [E] to the rank of E
• The tensor product of vector bundles induces a ring structure on K(X)
  • The product is given by [E] · [F] = [E ⊗ F]
• The exterior powers of vector bundles induce operations on K(X) called lambda operations
  • The kth lambda operation is defined by λk([E]) = [ΛkE], where ΛkE is the kth exterior power of E

K-Theory and Vector Bundles

Relationship between K(X) and Vector Bundles

• The monoid of isomorphism classes of complex vector bundles over X, denoted Vect(X), forms a monoid under the direct sum operation
• K(X) is defined as the Grothendieck group of Vect(X), i.e., K(X) = G(Vect(X))
  • The Grothendieck group of a monoid M, denoted G(M), is the group completion of M, obtained by formally adding inverses for each element of M
• The map Vect(X) → K(X) sending a vector bundle to its isomorphism class is a monoid homomorphism
  • This map induces the universal map from Vect(X) to its Grothendieck group

Grothendieck Group Construction

• The Grothendieck group construction allows us to transform a monoid (Vect(X)) into a group (K(X))
• Formally, the Grothendieck group G(M) of a monoid M is the quotient of M × M by the equivalence relation (a, b) ~ (c, d) if there exists e ∈ M such that a + d + e = b + c + e
  • The equivalence class of (a, b) is denoted [a - b] and represents the "difference" between a and b
• The group operation in G(M) is given by [a - b] + [c - d] = [a + c - (b + d)]
• The monoid homomorphism M → G(M) sends a to [a - 0], where 0 is the identity element of M

Properties of K-Theory

Functoriality

• If f: X → Y is a continuous map between compact Hausdorff spaces, then there is an induced group homomorphism f*: K(Y) → K(X)
  • f* is defined by f*([E]) = [f*(E)], where f*(E) is the pullback of the vector bundle E along f
• The pullback operation on vector bundles is compatible with direct sums and preserves isomorphism classes
  • This ensures that f* is a well-defined group homomorphism

Homotopy Invariance

• If f, g: X → Y are homotopic continuous maps between compact Hausdorff spaces, then the induced maps f*, g*: K(Y) → K(X) are equal
• Homotopic maps induce isomorphic pullback bundles
  • This fact, combined with the definition of K(X) in terms of isomorphism classes of vector bundles, implies homotopy invariance
• Homotopy invariance allows us to define K-theory for homotopy types of spaces
  • If X and Y are homotopy equivalent, then K(X) and K(Y) are isomorphic

K-Theory for Simple Spaces

K-Theory of a Point

• For a one-point space , K() is isomorphic to the integers Z
• The isomorphism K(*) → Z is given by the rank of vector bundles
  • Every vector bundle over a point is trivial and characterized by its rank
• The inverse isomorphism Z → K(*) sends n to the isomorphism class of the trivial bundle of rank n

K-Theory of Spheres

• For the n-sphere Sn, K(Sn) is isomorphic to Z for n even and Z ⊕ Z for n odd
• For even n, the isomorphism K(Sn) → Z is given by the rank of vector bundles
• For odd n, the isomorphism K(Sn) → Z ⊕ Z is given by the rank and the "reduced" rank
  • The reduced rank measures the non-triviality of the vector bundle
  • It is defined as the difference between the rank of the bundle and the rank of its restriction to the equator Sn-1
• The computation of K(Sn) relies on the clutching construction
  • The clutching construction builds vector bundles over Sn from transition functions on the equator Sn-1
• Vector bundles over contractible spaces (such as the northern and southern hemispheres of Sn) are trivial
  • This fact is used in the computation of K(Sn)