2.4 Reduced K-Theory and suspension isomorphism

Reduced K-theory and the suspension isomorphism are key concepts in K-theory, building on the Grothendieck group. They provide powerful tools for computing K-groups of spaces and understanding their structure. These ideas connect vector bundles to homotopy theory, laying groundwork for deeper results.

The suspension isomorphism relates K-groups of a space to those of its suspension, allowing for iterative calculations. This leads to periodicity in K-theory and motivates the development of higher K-theory, extending these ideas to more general algebraic settings.

Reduced K-theory

Definition and Relationship to Unreduced K-theory

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• Reduced K-theory $\tilde{K}(X)$ is defined for a pointed compact Hausdorff space $X$ as the kernel of the map $K(X) \to K(*)$ induced by the map $X \to *$ collapsing $X$ to a point
• A natural split short exact sequence $0 \to \tilde{K}(X) \to K(X) \to K(*) \to 0$ allows one to recover $K(X)$ from $\tilde{K}(X)$ and the rank of vector bundles
• For a non-compact space $X$, reduced K-theory can be defined as $\tilde{K}(X) = \ker(K(X^+) \to K(*))$, where $X^+$ is the one-point compactification of $X$
• Reduced K-theory is a contravariant functor from the category of pointed compact Hausdorff spaces to the category of abelian groups
• The reduced K-theory of a point is trivial, i.e., $\tilde{K}(*) = 0$, which distinguishes it from unreduced K-theory ($K(*)=\mathbb{Z}$)

Properties and Functoriality

• Reduced K-theory is a contravariant functor from the category of pointed compact Hausdorff spaces to the category of abelian groups
  • For a pointed map $f: X \to Y$, there is an induced homomorphism $\tilde{K}(f): \tilde{K}(Y) \to \tilde{K}(X)$
  • Functoriality means that $\tilde{K}(\mathrm{id}_X) = \mathrm{id}_{\tilde{K}(X)}$ and $\tilde{K}(g \circ f) = \tilde{K}(f) \circ \tilde{K}(g)$ for pointed maps $f: X \to Y$ and $g: Y \to Z$
• The reduced K-theory of a point is trivial, i.e., $\tilde{K}(*) = 0$, which is the key property that distinguishes it from unreduced K-theory
  • This property allows for the construction of the split exact sequence relating reduced and unreduced K-theory
  • It also implies that reduced K-theory is a reduced cohomology theory, satisfying $\tilde{K}(X \vee Y) \cong \tilde{K}(X) \oplus \tilde{K}(Y)$ for pointed spaces $X$ and $Y$
• Reduced K-theory is related to unreduced K-theory via the split exact sequence $0 \to \tilde{K}(X) \to K(X) \to K(*) \to 0$
  • This sequence allows for the computation of $K(X)$ from $\tilde{K}(X)$ and the rank of vector bundles over $X$
  • The splitting of the sequence is given by the map $K(*) \to K(X)$ induced by the inclusion of a point into $X$

Suspension Isomorphism for Reduced K-theory

Statement and Naturality

• The suspension isomorphism theorem states that for a pointed compact Hausdorff space $X$, there is a natural isomorphism $\tilde{K}(\Sigma X) \cong \tilde{K}(X)$, where $\Sigma X$ is the reduced suspension of $X$
  • The reduced suspension $\Sigma X$ is the quotient space $(X \times I)/(X \times \{0, 1\} \cup {*} \times I)$, where $I = [0, 1]$ and $*$ is the basepoint of $X$
  • The suspension isomorphism relates the reduced K-theory of a space to that of its suspension, providing a powerful computational tool
• The suspension isomorphism is natural with respect to pointed maps, i.e., for a pointed map $f: X \to Y$, the following diagram commutes: \begin{CD} \tilde{K}(\Sigma X) @>{\tilde{K}(\Sigma f)}>> \tilde{K}(\Sigma Y) \\ @V{\cong}VV @VV{\cong}V \\ \tilde{K}(X) @>>{\tilde{K}(f)}> \tilde{K}(Y) \end{CD}
  • Naturality means that the suspension isomorphism is compatible with the functorial behavior of reduced K-theory
  • This property is crucial for the application of the suspension isomorphism in computations and the development of higher K-theory

Proof Sketch and Key Ideas

• The proof of the suspension isomorphism involves constructing explicit maps $\tilde{K}(\Sigma X) \to \tilde{K}(X)$ and $\tilde{K}(X) \to \tilde{K}(\Sigma X)$ and showing that they are inverses
  • The map $\tilde{K}(\Sigma X) \to \tilde{K}(X)$ is constructed using the clutching function of a vector bundle over $\Sigma X$
  • The map $\tilde{K}(X) \to \tilde{K}(\Sigma X)$ is constructed by gluing together trivial bundles over the two cones of $\Sigma X$ using a vector bundle over $X$
• The key idea is that vector bundles over the suspension $\Sigma X$ can be constructed by gluing together trivial bundles over the two cones of $\Sigma X$ along $X$
  • This gluing data is precisely a vector bundle over $X$, establishing a correspondence between vector bundles over $\Sigma X$ and those over $X$
  • The clutching construction makes this correspondence explicit and allows for the definition of the isomorphism
• The suspension isomorphism is a key tool in computing reduced K-theory groups and underlies the Bott periodicity theorem
  • Bott periodicity states that there are natural isomorphisms $\tilde{K}(\Sigma^2 X) \cong \tilde{K}(X)$ for all pointed compact Hausdorff spaces $X$
  • The suspension isomorphism is the key ingredient in the proof of Bott periodicity and its generalizations to higher K-theory

Reduced K-theory of Suspensions

Iterating the Suspension Isomorphism

• For a pointed compact Hausdorff space $X$, the suspension isomorphism allows one to compute $\tilde{K}(\Sigma X)$ in terms of $\tilde{K}(X)$
  • This process can be iterated to compute the reduced K-theory of higher suspensions of $X$
  • For example, $\tilde{K}(\Sigma^2 X) \cong \tilde{K}(\Sigma X) \cong \tilde{K}(X)$, where $\Sigma^2 X = \Sigma(\Sigma X)$ is the double suspension of $X$
• Iterating the suspension isomorphism, one obtains isomorphisms $\tilde{K}(\Sigma^n X) \cong \tilde{K}(X)$ for all $n \geq 0$, where $\Sigma^n X$ is the $n$-fold reduced suspension of $X$
  • This allows for the computation of the reduced K-theory of iterated suspensions of a space in terms of the reduced K-theory of the original space
  • These isomorphisms are natural with respect to pointed maps, i.e., they are compatible with the functorial behavior of reduced K-theory
• The suspension isomorphism is also compatible with the external product in K-theory, allowing for the computation of the reduced K-theory of smash products of spaces
  • For pointed spaces $X$ and $Y$, there is a natural isomorphism $\tilde{K}(X \wedge Y) \cong \tilde{K}(X) \otimes \tilde{K}(Y)$, where $\wedge$ denotes the smash product
  • Combined with the suspension isomorphism, this allows for the computation of the reduced K-theory of suspensions and smash products of spaces

Examples and Computations

• For the $n$-sphere $S^n$, the reduced K-theory is given by $\tilde{K}(S^n) \cong \mathbb{Z}$ for $n$ even and $\tilde{K}(S^n) \cong 0$ for $n$ odd
  • This computation follows from the suspension isomorphism and the fact that $S^n \cong \Sigma S^{n-1}$ for $n \geq 1$
  • For example, $\tilde{K}(S^2) \cong \tilde{K}(\Sigma S^1) \cong \tilde{K}(S^1) \cong \mathbb{Z}$ and $\tilde{K}(S^3) \cong \tilde{K}(\Sigma S^2) \cong \tilde{K}(S^2) \cong 0$
• More generally, for a pointed CW complex $X$, the reduced K-theory of $X$ can be computed using the Atiyah-Hirzebruch spectral sequence
  • This spectral sequence relates the reduced K-theory of $X$ to its cellular homology groups
  • The suspension isomorphism is a key ingredient in the construction of this spectral sequence and in the computation of differentials
• The reduced K-theory of the complex projective space $\mathbb{C}P^n$ is given by $\tilde{K}(\mathbb{C}P^n) \cong \mathbb{Z}^n$
  • This computation can be performed using the cellular structure of $\mathbb{C}P^n$ and the Atiyah-Hirzebruch spectral sequence
  • Alternatively, it can be deduced from the cofiber sequence $S^1 \to S^{2n+1} \to \mathbb{C}P^n$ and the associated long exact sequence in reduced K-theory

Reduced K-theory in Higher K-theory

Motivation and Analogies

• The development of higher K-theory, which associates K-groups $K_n(R)$ to a ring $R$ for all integers $n$, is motivated by the properties of reduced K-theory
  • The success of topological K-theory in solving geometric problems, such as the vector fields on spheres problem, inspired the search for algebraic analogues of K-theory
  • The functorial properties and computations in reduced K-theory serve as a guide for the development of higher algebraic K-theory
• The suspension isomorphism in reduced K-theory suggests that there should be a periodic family of K-groups $K_n(X)$ associated to a space $X$, with $K_0(X) = K(X)$ and $K_{-1}(X) = \tilde{K}(X)$
  • This idea leads to the Bott periodicity theorem, which states that there are natural isomorphisms $K_n(X) \cong K_{n+2}(X)$ for all $n$
  • Bott periodicity can be viewed as an infinite iteration of the suspension isomorphism, extending the periodicity of reduced K-theory to all higher K-groups
• The functorial properties of reduced K-theory, such as the long exact sequence associated to a pair $(X, A)$, motivate the development of analogous long exact sequences in higher K-theory
  • For a ring $R$ and an ideal $I$, there is a long exact sequence relating the higher K-groups of $R$, $I$, and $R/I$
  • This long exact sequence is a key tool in the computation of higher algebraic K-groups and the study of the K-theory of rings

Constructions and Results

• The construction of higher algebraic K-theory by Quillen and others builds upon the ideas and techniques of topological K-theory
  • Quillen's plus construction associates to a ring $R$ a topological space $BGL(R)^+$, whose homotopy groups are the higher K-groups of $R$
  • This construction is analogous to the classifying space of vector bundles in topological K-theory and allows for the application of homotopy-theoretic techniques
• The Bott periodicity theorem in higher algebraic K-theory states that for a regular noetherian ring $R$, there are natural isomorphisms $K_n(R) \cong K_{n+2}(R)$ for all $n$
  • This theorem extends the periodicity of reduced K-theory to higher algebraic K-theory
  • The proof of Bott periodicity in this setting relies on the properties of the algebraic K-theory spectrum and the use of trace methods
• The Quillen-Lichtenbaum conjecture relates the higher algebraic K-theory of a ring of integers in a number field to its étale cohomology groups
  • This conjecture provides a link between algebraic K-theory and arithmetic geometry, analogous to the connection between topological K-theory and complex geometry
  • The proof of the Quillen-Lichtenbaum conjecture for regular schemes relies on the development of motivic cohomology and the use of spectral sequences
• The higher algebraic K-theory of fields and local rings has important applications in algebraic geometry and number theory
  • The K-groups of fields are related to the Milnor K-theory and the study of Galois cohomology
  • The K-groups of local rings are related to the study of singularities and the resolution of curve singularities in algebraic geometry