2.3 Functorial properties of K-Theory

K-Theory's functorial properties reveal its deep connection to topology. As a contravariant functor, it maps spaces to groups, reversing morphisms. This powerful tool lets us analyze complex spaces by breaking them down into simpler parts.

Understanding how K-Theory behaves with products and quotients is crucial. It allows us to compute K-groups of intricate spaces, like tori or projective planes, by relating them to simpler spaces we already know.

Functorial properties of K-theory

K-theory as a contravariant functor

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• K-theory is a contravariant functor from the category of compact Hausdorff spaces to the category of abelian groups
  • Reverses the direction of morphisms and preserves composition
  • A continuous map $f: X \to Y$ between compact Hausdorff spaces induces a homomorphism $f^*: K(Y) \to K(X)$ between their K-theory groups, going in the opposite direction
  • The functoriality of K-theory implies that the identity map on a space induces the identity homomorphism on its K-theory group, and the composition of continuous maps induces the composition of the corresponding homomorphisms in the opposite order
  • Example: If $f: X \to Y$ and $g: Y \to Z$ are continuous maps, then $(f \circ g)^* = g^* \circ f^*$

Proving K-theory is a contravariant functor

• To prove that K-theory is a contravariant functor, one needs to show that it reverses the direction of morphisms and preserves composition
  • Given a continuous map $f: X \to Y$ between compact Hausdorff spaces, the induced homomorphism $f^*: K(Y) \to K(X)$ is defined by pulling back vector bundles
  • The proof involves showing that the identity map on a space induces the identity homomorphism on its K-theory group, i.e., $(id_X)^* = id_{K(X)}$
  • The proof also requires demonstrating that the composition of continuous maps induces the composition of the corresponding homomorphisms in the opposite order, i.e., $(f \circ g)^* = g^* \circ f^*$ for continuous maps $f: X \to Y$ and $g: Y \to Z$
  • Example: Consider the identity map $id_X: X \to X$ and a continuous map $f: X \to Y$. The proof would involve showing that $(id_X)^* = id_{K(X)}$ and $(f \circ id_X)^* = (id_X)^* \circ f^*$

Induced maps in K-theory

Constructing induced maps

• Given a continuous map $f: X \to Y$ between compact Hausdorff spaces, the induced homomorphism $f^*: K(Y) \to K(X)$ is constructed by pulling back vector bundles
  • The pullback of a vector bundle $E$ over $Y$ along the map $f$ is a vector bundle $f^*(E)$ over $X$, whose fiber over a point $x \in X$ is the same as the fiber of $E$ over $f(x) \in Y$
  • The pullback operation is compatible with the Whitney sum and tensor product of vector bundles, i.e., $f^*(E \oplus F) \cong f^*(E) \oplus f^*(F)$ and $f^*(E \otimes F) \cong f^*(E) \otimes f^*(F)$
  • The induced homomorphism $f^*$ maps the equivalence class $[E] - [F]$ in $K(Y)$ to the equivalence class $[f^*(E)] - [f^*(F)]$ in $K(X)$, where $E$ and $F$ are vector bundles over $Y$
  • Example: Let $f: S^1 \to S^1$ be the map that sends $z$ to $z^2$. The pullback of the Möbius bundle over $S^1$ along $f$ is the trivial bundle over $S^1$

K-theory of products vs quotients

K-theory of product spaces

• The functorial properties of K-theory can be used to compute the K-theory groups of product spaces in terms of the K-theory groups of their constituent spaces
  • For a product space $X \times Y$, the projection maps $\pi_X: X \times Y \to X$ and $\pi_Y: X \times Y \to Y$ induce homomorphisms $\pi_X^*: K(X) \to K(X \times Y)$ and $\pi_Y^*: K(Y) \to K(X \times Y)$, which together yield an isomorphism $K(X) \oplus K(Y) \cong K(X \times Y)$
  • Example: The K-theory group of the torus $S^1 \times S^1$ is isomorphic to $K(S^1) \oplus K(S^1) \cong \mathbb{Z} \oplus \mathbb{Z}$

K-theory of quotient spaces

• The functorial properties of K-theory can be used to compute the K-theory groups of quotient spaces in terms of the K-theory groups of their constituent spaces
  • For a quotient space $X/A$, where $A$ is a closed subspace of $X$, the quotient map $q: X \to X/A$ induces a homomorphism $q^*: K(X/A) \to K(X)$, and the inclusion map $i: A \to X$ induces a homomorphism $i^*: K(X) \to K(A)$
  • These homomorphisms fit into a long exact sequence relating the K-theory groups of $X$, $A$, and $X/A$
  • The functorial properties can be applied iteratively to compute the K-theory groups of more complex spaces built from simpler ones using quotients
  • Example: The K-theory group of the real projective plane $\mathbb{RP}^2$, which is the quotient of the 2-sphere $S^2$ by the antipodal map, can be computed using the long exact sequence associated with the quotient map $S^2 \to \mathbb{RP}^2$