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S 1 -Equivariant K-Theory of CP1 Daniel Hudson University of Victoria [email protected] April 20, 1969 G -Spaces Definition If G is a topological group and X is a topological space, then we say X is a G -space if is equipped with a continuous action G × X → X. G -Spaces Definition If G is a topological group and X is a topological space, then we say X is a G -space if is equipped with a continuous action G × X → X. Example. Rn is naturally a GLn (R)-space with the action (A, x) 7→ Ax. G -Spaces Definition If G is a topological group and X is a topological space, then we say X is a G -space if is equipped with a continuous action G × X → X. Example. Rn is naturally a GLn (R)-space with the action (A, x) 7→ Ax. Example. CP1 , the space C2 modulo the relation x ∼ y if y = λx for λ ∈ C, is naturally a S 1 -space via the action z 0 (z, [d]) 7→ d . 0 z̄ Vector Bundles Definition If X is a G -space and {Ex : x ∈ X } is a collection of finite dimensional vector spaces, we call E = tx∈X Ex a G -vector bundle if it is equipped with certain topological structure and a continuous action of G such that g .Ex = Eg .x . Vector Bundles Definition If X is a G -space and {Ex : x ∈ X } is a collection of finite dimensional vector spaces, we call E = tx∈X Ex a G -vector bundle if it is equipped with certain topological structure and a continuous action of G such that g .Ex = Eg .x . Example. (The Hopf Bundle) A canonical example of a S 1 -bundle over CP1 is H ∗ := {([d], z) ∈ CP1 × C2 : z is a point in the line [d]}. We define the Hopf Bundle to be the dual bundle, ∗ ∗ H := t[d]∈CP1 (H[d] ) . Construction of KG (X ) We can use G -vector bundles over compact Hausdorff G -spaces X to form a homotopy invariant cohomology theory KG (X ) as follows. Construction of KG (X ) We can use G -vector bundles over compact Hausdorff G -spaces X to form a homotopy invariant cohomology theory KG (X ) as follows. KG (X ) • Define E ⊕ F and E ⊗ F such that (E ⊕ F )x = Ex ⊕ Fx and (E ⊗ F )x = Ex ⊗ Fx ; Construction of KG (X ) We can use G -vector bundles over compact Hausdorff G -spaces X to form a homotopy invariant cohomology theory KG (X ) as follows. KG (X ) • Define E ⊕ F and E ⊗ F such that (E ⊕ F )x = Ex ⊕ Fx and (E ⊗ F )x = Ex ⊗ Fx ; • (VectG (X ), ⊕) is an abelian semi-group with unity; Construction of KG (X ) We can use G -vector bundles over compact Hausdorff G -spaces X to form a homotopy invariant cohomology theory KG (X ) as follows. KG (X ) • Define E ⊕ F and E ⊗ F such that (E ⊕ F )x = Ex ⊕ Fx and (E ⊗ F )x = Ex ⊗ Fx ; • (VectG (X ), ⊕) is an abelian semi-group with unity; • Define KG (X ) := G(VectG (X )), where G(VectG (X )) is the “Grothendieck completion” of VectG (X ). The tensor product gives it the structure of a commutative ring. Construction of KG (X ) We can use G -vector bundles over compact Hausdorff G -spaces X to form a homotopy invariant cohomology theory KG (X ) as follows. KG (X ) • Define E ⊕ F and E ⊗ F such that (E ⊕ F )x = Ex ⊕ Fx and (E ⊗ F )x = Ex ⊗ Fx ; • (VectG (X ), ⊕) is an abelian semi-group with unity; • Define KG (X ) := G(VectG (X )), where G(VectG (X )) is the “Grothendieck completion” of VectG (X ). The tensor product gives it the structure of a commutative ring. Example. KG (point) = R(G ), the representation ring of G , which is defined as the Grothendieck completion of equivalence classes of representations of G . Properties of KG (X ) If f : X → Y is a continuous map which commutes with the action of G , then one can define a map f ∗ : KG (Y ) → KG (X ). Properties of KG (X ) If f : X → Y is a continuous map which commutes with the action of G , then one can define a map f ∗ : KG (Y ) → KG (X ). If f is a homotopy equivalence, then f ∗ is a group isomorphism, and id∗X = idKG (X ) . Properties of KG (X ) If f : X → Y is a continuous map which commutes with the action of G , then one can define a map f ∗ : KG (Y ) → KG (X ). If f is a homotopy equivalence, then f ∗ is a group isomorphism, and id∗X = idKG (X ) . Thus, KG is a homotopy invariant contravariant functor from the category of compact Hausdorff G -spaces to the category of abelian groups. For any G -space X there is a natural map X → {∗}. By the contravariance property, this allows us to interpret KG (X ) as an algebra over R(G ) ∼ = KG (∗). Goal Compute KS∗1 (CP1 ) as an algebra over R(S 1 ). Resolution • If follows from a theorem of Segal that KS∗1 (CP1 ) is generated by the Hopf H as an algebra of R(S 1 ) modulo the P bundle k k relation k (−1) Λ C2 .H k = 0. Resolution • If follows from a theorem of Segal that KS∗1 (CP1 ) is generated by the Hopf H as an algebra of R(S 1 ) modulo the P bundle k k relation k (−1) Λ C2 .H k = 0. • Using basic representation theory, one can show that R(S 1 ) ∼ = Z[X , X −1 ]. Resolution • If follows from a theorem of Segal that KS∗1 (CP1 ) is generated by the Hopf H as an algebra of R(S 1 ) modulo the P bundle k k relation k (−1) Λ C2 .H k = 0. • Using basic representation theory, one can show that R(S 1 ) ∼ = Z[X , X −1 ]. • One can write this isomorphism down explicitly to see that Λ0 C2 = Λ2 C2 = 1, Λ1 C2 = X + X −1 Λk C2 = 0 for k > 2. Resolution • If follows from a theorem of Segal that KS∗1 (CP1 ) is generated by the Hopf H as an algebra of R(S 1 ) modulo the P bundle k k relation k (−1) Λ C2 .H k = 0. • Using basic representation theory, one can show that R(S 1 ) ∼ = Z[X , X −1 ]. • One can write this isomorphism down explicitly to see that Λ0 C2 = Λ2 C2 = 1, Λ1 C2 = X + X −1 Λk C2 = 0 for k > 2. Putting this all together, we have: Conclusion Theorem Let S 1 act on CP1 by (z, [d]) 7→ z 0 d . 0 z̄ Then KS∗1 (CP1 ) is generated as an algebra over R(S 1 ) ∼ = Z[X , X −1 ] by the Hopf bundle H and 1 subject to the relation H 2 = (X + X −1 )H − 1. Further Reading [1] M. F. Atiyah and D. W. Anderson. K-Theory. W.A. Benjamin, Inc, 1967. [2] G. Segal, Equivariant K-theory, Publications Mathematiques de lI.H. E.S., tome 34 (1968), p.129 - 151. Thanks for listening!