Download S1-Equivariant K-Theory of CP1

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!