11.3 Bott periodicity and the six-term exact sequence

Bott periodicity reveals a repeating pattern in K-theory groups every two dimensions. This deep connection between different dimensions provides powerful tools for calculating K-groups of various spaces.

The six-term exact sequence is a key tool in K-theory for analyzing short exact sequences of C*-algebras. It connects K0 and K1 groups of three algebras, allowing us to compute unknown K-groups when others are known.

Bott Periodicity and Suspension

Fundamental Concepts of Bott Periodicity

• Bott periodicity theorem establishes a deep connection between K-theory groups of different dimensions
• Reveals a repeating pattern in K-theory groups every two dimensions
• Applies to both real and complex K-theory, with different periodicities (8 for real, 2 for complex)
• Fundamental result in topology and K-theory, originally discovered by Raoul Bott in the late 1950s
• Provides powerful computational tools for calculating K-groups of various spaces

Suspension and Its Role in K-Theory

• Suspension of C*-algebras involves adding a dimension to the algebra
• Defined as $SA = C_0((0,1)) \otimes A$ where $C_0((0,1))$ represents continuous functions vanishing at endpoints
• Suspension operation crucial for understanding K-theory of higher dimensions
• Relates K-groups of an algebra to those of its suspension through $K_1(A) \cong K_0(SA)$
• Allows for iterative computation of K-groups using suspension isomorphisms

Exponential Map and Its Significance

• Exponential map connects unitaries in a C*-algebra to its K-theory
• Defined as $\exp: M_n(A) \to U_n(A)$ where $M_n(A)$ represents n×n matrices over A
• Plays a crucial role in defining K-theory groups and establishing isomorphisms
• Facilitates the transition between K0 and K1 groups in the Bott periodicity theorem
• Used to construct explicit isomorphisms in the proof of Bott periodicity

Six-Term Exact Sequence

Structure and Components of the Six-Term Sequence

• Six-term exact sequence fundamental tool in K-theory for analyzing short exact sequences of C*-algebras
• Consists of K0 and K1 groups of three C*-algebras involved in a short exact sequence
• Takes the form: $K_0(A) \to K_0(B) \to K_0(C)$ $\uparrow \qquad \qquad \qquad \downarrow$ $K_1(C) \leftarrow K_1(B) \leftarrow K_1(A)$
• Arrows represent group homomorphisms induced by the maps in the original short exact sequence
• Exactness implies the image of each map equals the kernel of the next map in the sequence

Boundary Maps and Their Significance

• Boundary maps connect K0 and K1 groups in the six-term exact sequence
• Defined as $\partial_0: K_0(C) \to K_1(A)$ and $\partial_1: K_1(C) \to K_0(A)$
• Arise from the connecting homomorphisms in algebraic topology
• Crucial for understanding how K-theory information passes between different algebras in the sequence
• Often challenging to compute explicitly, but provide valuable insights into the structure of K-groups

Index Map and Its Applications

• Index map relates to the boundary map $\partial_1: K_1(C) \to K_0(A)$ in certain contexts
• Assigns an integer (index) to certain operators, providing a link between analysis and topology
• Plays a central role in the Atiyah-Singer Index Theorem, connecting analytical and topological invariants
• Used in various areas of mathematics and physics, including gauge theory and string theory
• Provides a method for computing K-theory groups in specific situations

Long Exact Sequence and Its Implications

• Six-term exact sequence part of a more general long exact sequence in K-theory
• Can be extended indefinitely by applying suspension repeatedly
• Takes the form: $\cdots \to K_0(A) \to K_0(B) \to K_0(C) \to K_1(A) \to K_1(B) \to K_1(C) \to K_0(A) \to \cdots$
• Allows for computation of K-groups of one algebra if the K-groups of the other two are known
• Demonstrates the deep interplay between different dimensions in K-theory, reflecting Bott periodicity