K-Theory for Locally Compact Spaces Airi Takeuchi (Keio University)

Definition of Vect(X ) The Six-Term Exact Sequence

Let X be a topological space. Let A be a closed subspace of a locally compact space X , and let • The collection of isomorphism classes hξi of complex vector bundles j : A → X be the inclusion map. Then there exists an exact sequence 0 0 j∗ 0 ξ over X is denoted Vect(X ). K (X \A) / K (X ) / K (A) O • Addition on Vect(X ) is defined by hξi + hηi = hξ ⊕ ηi, where δ δ −1 j∗ −1 −1  ξ ⊕ η := {(v1, v2) ∈ V1 × V2 | π1(v1) = π2(v2)} is the direct sum of K (A) o K (X ) o K (X \A) two vector bundles ξ = (V1, π1) and η = (V2, π2). with natural connecting maps δ.

Definition of K −n(X ) Definition of RPn Pn is defined as the quotient space S n/ ∼ , where • Let X be a compact Hausdorff space. Define K 0(X ) be the R Grothendieck group of (Vect(X ), +). x ∼ y ⇐⇒ x = ±y, equipped with the quotient topology.

+  Remark  • Let X be a locally compact Hausdorff Space and let X be the n−1 n 0 • For each natural number n, RP is a closed subset of RP one-point compactification of X . Define K (X ) be n n−1 ∼ n ∗ 0 + 0 ∼ and P \ P = Ker(j : K (X ) → K (pt) = Z), where the group homomorphism R R R j ∗ is induced by the inclusion map j : pt → X +.   −n 0 n • Define K (X ) = K (X × R ) for every natural number n and K-Groups of RPn locally compact Hausdorff space X . Consider the following commutative diagram  Remark  0 n n−1 • The definition of K -groups for locally compact spaces is an o ? _ RPO RPO extension of that for compact spaces. n n−1 S o ? _S .   This yields The Long Exact Sequence 0 n 0 n 0 n−1 K (R ) / K (RP ) / K (RP ) O

Let A be a closed subspace of a locally compact Hausdorff space X , 0 n v n 0 n x 0 n−1 v K (R q R ) / K (S ) / K (S ) δ and let j : A → X be the inclusion map. Then, for each natural number O − − −  n there exists a natural homomorphism δ : K n 1(X ) → K n(X \A) −1 n−1 −1 n δ −1 n δ K (RP ) o K (RP ) o K (R ) that makes the infinite sequence v x  v j∗ δ −1 n−1 −1 n −1 n n · · · → K −2(X ) −→ K −2(A) −→ K (S ) o K (S ) o K (R q R ) j∗ δ j∗ The naturality of the connecting maps δ makes the above diagram K −1(X \A) → K −1(X ) −→ K −1(A) −→ K 0(X \A) → K 0(X ) −→ K 0(A) commutative. exact. By the diagram chase, inductively we obtain ( n ⊕ ( /2 )2 (if n is even) Bott Periodicity 0 n ∼ Z Z Z K (RP ) = n−1 Z ⊕ (Z/2Z) 2 (if n is odd) For every locally compact Hausdorff space X , there exists a natural and isomorphism ( −1 n ∼ 0 (if n is even) β : K 0(X ) → K −2(X ). K (RP ) = Z (if n is odd).

K-Groups of Rn and S n Definition of CPn

 Fact  n n+1 0 −1 P is defined as the quotient space ( \{0})/ ∼, where • K (pt) ∼= and K (pt) ∼= 0. C C Z x ∼ y ⇐⇒ x = λy for some complex number λ, equipped with the   Bott periodicity gives us quotient topology.

(  Remark  if n is even n−1 n K 0( n) ∼= K 0((pt) × n) = K −n(pt) ∼= Z • For each natural number n, CP is a closed subset of CP R R n n−1 ∼ n 0 if n is odd. and CP \CP = C This yields   ( 0 if n is even K-Groups of CPn K −1( n) = K 0( n+1) ∼= R R if n is odd. Z Considering the inclusion n n−1  Fact  CP ←- CP , • Let X + be the one-point compactification of a locally we have the exact sequence compact space X , then K 0(X +) ∼= K 0(X ) ⊕ and Z 0 n 0 n 0 n−1 −1 + ∼ −1 K (C ) / K (CP ) / K (CP ) K (X ) = K (X ). O   −1 n−1 −1 n −1  n K ( P ) o K ( P ) o K ( ). Since S n is the one-point compactification of Rn, we obtain C C C ( 2 This yields 0 n ∼ 0 n + ∼ 0 n ∼ Z if n is even 0 n ∼ n+1 K (S ) = K ((R ) ) = K (R ) ⊕ Z = K (CP ) = Z Z if n is odd. and −1 n ∼ Similarly, K (CP ) = 0. ( −1 n ∼ −1 n + ∼ −1 n ∼ 0 if n is even K (S ) = K ((R ) ) = K (R ) = References Z if n is odd. • Park, Efton. Complex topological K-theory. Vol. 111. Cambridge University Press, 2008. • Rørdam, Mikael, Flemming Larsen, and Niels Laustsen. An Introduction to K-theory for C*-algebras. Vol. 49. Cambridge University Press, 2000.