Equivariant K-Theory of CP
1
This research was supported by the Jamie Cassels
Undergraduate Research Award,
Supervised by Dr. Heath Emerson,
Department of Mathematics and Statistics,
University of Victoria
Daniel R. H. Hudson
{drhh}@uvic.ca
Historical Background
G-Vector Bundles
Algebraic K-theory was invented by Alexander Grothendieck to generalize the
Riemann-Roch theorem.
• Some spaces can be naturally equipped with multiplication by a group, called a
group action.
Sir Michael Francis Atiyah invented topological K-theory in order to formulate
and prove the Atiyah-Singer Index Theorem, one of the most important theorems
of the past century which shows a deep connection between analysis and topology.
• A space X with an action of a group G is called a G-space.
Equivariant K-theory was developed by Graeme Segal who introduced the additional structure of a group action.
• A G-family of vector spaces E over a G-space X is a collection of vector spaces
Ex indexed by each point x ∈ X. In addition, there exist linear maps Ex → Eg.x
for each g ∈ G, x ∈ X.
• Families of vector spaces over a G-space X are also G-spaces.
• A G-vector bundle E over a G-space X is a family of vector spaces over X such
that E locally looks like U × Cn, for some n and some open U ⊂ X.
Example 3.
• A canonical example of a G-vector bundle over CP1
is
Figure 2: Sir Micheal Francis
Atiyah, (1)
Figure 3: Graeme Segal, (3)
H ∗ := {(L, z) ∈ CP × C2 : z is a point in the line L}.
• The Hopf Bundle H is defined to be the dual of H ∗.
Example 4.
Mathematical Background
• The Möbius Bundle is shown in Figure 6.
• A topological space is a space in which we have some notion of geometry.
• A topological space is compact if it satisfies a certain “finiteness” condition.
• All topological spaces here will be compact.
Equivariant K-Theory
KG∗ (X) forms an algebra over R(G) by functoriality. We are interested in studying
1
∗
the structure of KG(CP ) as an algebra over R(G). To do this, we need a theorem
from [4].
Theorem 1. [4] Let G be a compact group. If E is a G-vector bundle on X then
∗
(X)-algebra
by
the
Hopf
Bundle
H,
modulo
the
is
generated
as
a
K
KG∗ (P (E))
G
P
relation k (−1)k Λk E.H k = 0.
Let G = T, the circle. We let T act on CP1 by
z0
(z, L) 7→
L.
0 z̄
Figure 6: The Möbius Bundle, (4)
• Topological spaces X and Y are homotopy equivalent if they can be stretched
and contracted into one another.
Summarizing, one has:
Theorem 2. Let T act on CP1 by
(z, L) 7→
• We like to associate algebraic structures, such as a ring, to topological spaces
X and Y such that if X and Y are homotopy equivalent, then they have the same
algebraic structure. Such an association is called a homotopy invariant.
Example 1.
n
• An important example is the n-torus, T .
• The n-torus is the product of n-circles.
• A compact topological group.
Figure 4: The 2-torus, T2, (5)
• The equivariant K-theory of a G-space X, denoted KG(X), is the so-called
Grothendieck completion of the abelian semi-group of isomorphism classes of
vector bundles with the direct sum. Using the tensor product to define a multiplication gives KG(X) the structure of a commutative ring.
Example 5.
Example 2.
• Main example is the Complex Projective Space, CPn.
z0
L.
0 z̄
H 2 = (X + X −1)H − 1.
References
[1] M. F. Atiyah, Lectures on K-theory, mimeographed, Harvard, 1964
[2] T. Bröcker, T. tom Dieck, Representations of Compact Lie Groups, Graduate Texts in Mathematics, 1985 Springer-Verlag New York Inc.
[3] H. Emerson, Localization Techniques in Circle-Equivariant Kasparov Theory, http://arxiv.org/pdf/1004.2970.pdf
[4] G. Segal, Equivariant K-theory, Publications Mathématiques de l’I.H.É.S., tome 34 (1968), p.129 - 151T. Bröcker, T. tom Dieck, Representations of Compact Lie Groups, Graduate Texts in Mathematics, 1985 Springer-Verlag New York Inc.
[5] G. Segal, The Representation-ring of a Compact Lie Group, Publications Mathématiques de l’I.H.É.S., tome 34 (1968), p.113 - 128
If G = I is the group with one element and X = {∗} is the one point space, then
KG(X) = Z, the integers.
[6] C. Teleman, Representation Theory Course Notes, https://math.berkeley.edu/ teleman/math/RepThry.pdf
Images
(1) By Gert-Martin Greuel (MFO: http://owpdb.mfo.de/detail?photoID=10118), via Wikimedia Commons
(2) By Konrad Jacobs, Erlangen, Copyright by MFO / Original uploader was AEDP at it.wikipedia - http://owpdb.mfo.de/detail?photoID=1452
n
• CP is the space of lines through the
origin in Cn+1.
Then KT∗ (CP1) is generated as an algebra over the Laurent polynomials Z[X, X −1]
by the Hopf bundle H and 1 subject to the relation
• We can build a homotopy invariant of this type using G-vector bundles, called
equivariant K-theory.
• Generalizes the shape of a doughnut.
Figure 5: The 1-dimensional complex projective space, (6)
Computation
1
∗
of KT(CP )
Using Theorem P
1, one can show that KT∗ (CP1) is generated by H as an R(T)algebra modulo k (−1)k Λk C2.H k = 0, which is less than illuminating. Using
basic representation theory of compact Lie groups, one can show that R(T) ∼
=
Z[X, X −1]. One can write this isomorphism down explicitly, and using this one
sees that the images of Λ0C2, Λ1C2, and Λ2C2 in Z[X, X −1] are 1, X + X −1,
k 2
and
1,
respectively.
Furthermore,
Λ
C = 0 for all k > 2, so the condition that
P
k k 2
k
2
−1
(−1)
Λ
C
.H
=
0
is
equivalent
to
saying
H
=
(X
+
X
)H − 1.
k
1
Figure 1:
Alexander
Grothendieck, (2)
More generally still, if X is a space which is homotopy equivalent to a point (i.e.
contractible), then KG(X) = R(G). This is because equivariant K-theory is a
homotopy invariant.
/ Transferred from it.wikipedia, CC BY-SA 2.0 de,
Figure 7: KI (∗) = Z, the number line above (7)
• A compact topological space.
https://commons.wikimedia.org/w/index.php?curid=3906524
(3) By George M. Bergman - http://owpdb.mfo.de/detail?photoID=5947
(4) Jakob.scholbach at en.wikipedia (http://www.gnu.org/copyleft/fdl.html)], from Wikimedia Commons
More generally, if G is any group and X is the one point space {∗}, then KG(X)
is the representation ring of G, denoted R(G).
(5) https://commons.wikimedia.org/w/index.php?curid=588734
(6) By Mark.Howison at English Wikipedia https://commons.wikimedia.org/w/index.php?curid=3380134
(7) By Hakunamenta - Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=20206520