Download Characteristic Classes

Characteristic Classes
by Caroline Junkins
January 25, 2012
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Vector Bundles over a Smooth Manifold
By smooth we will mean differentiable of class C ∞ , ie. for a function f defined on an open
set of Rn with values in Rk , f is smooth if all of its partial derivatives of all orders exist and
are continuous.
Let X be a topological space. A smooth real atlas on X is a collection of pairs (Ui , φi )
called coordinate charts, where the Ui are open sets which cover X and φi : Ui → Rn is
a homeomorphism of Ui onto an open subset of Rn . The maps φij := φj ◦ φ−1
|φi (Ui ∩Uj ) are
i
called transition maps (or gluing maps) and are smooth isomorphisms.
Two smooth atlases are called equivalent if their union is again a smooth atlas. An equivalence class of a smooth atlas is called a smooth structure on X.
A topological space X together with a smooth structure is called a smooth manifold,
and we say that n = dim(X).
• Euclidean space Rn is a smooth manifold of dimension n.
• The real n-sphere S n = {(x1 , . . . , xn+1 ) | x21 + · · · + x2n+1 = 1} is a smooth manifold of
dimension n.
• Real projective space RPn is a smooth manifold of dimension n.
• The real torus S 1 × S 1 is a smooth manifold of dimension 2.
Note that we may define other types of manifolds by altering our imposed conditions. For
instance, if we replace the base field by C, we obtain a complex manifold, or if we relax
the requirement on the transition functions to allow all continuous functions, this yields a
continuous manifold.
A real vector bundle of rank n over a smooth manifold X is a smooth manifold E together
with a smooth function η : E → X such that:
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• (fibration) For every point x ∈ X, the fiber Ex := η −1 (x) has the structure of an
n-dimensional real vector space (Rn ).
• (local triviality) For every x ∈ X there exists an open neighborhood U ⊂ X of x such
that η : η −1 (U ) → U is trivial. That is, there exists an siomorphism ψU : η −1 (U ) →
U × Rn .
• (compatibility) Let U, V ⊂ X be neighborhoods of x over which the bundle trivializes.
Then Rn → Fx → Rn is linear.
A homomorphism from one vector bundle η : E → X to another vector bundle φ : F →
X is a continuous map f : E → F with η = φf , such that each induced map fx : Ex → Fx
is a linear map of vector spaces.
f
Ex ⊆ E
F ⊇ Fx
η
φ
-
-
X
This allows us to say when two vector bundles are isomorphic, and so we may talk about
isomorphism class of vector bundles over X.
If f : Y → X is a continuous map, the we construct the pullback bundle f ∗ (η) : f ∗ E → Y
as follows:
Let f ∗ E = {(y, e) ∈ Y × E|f (y) = η(e)}. We equip f ∗ E with the subspace topology
and the projection map f ∗ (η) : f ∗ E → Y given by f ∗ (η)(y, e) = y. Let ρ : f ∗ E → E be
the projection onto the second factor: ρ(y, e) = e. Then we have the following commutative
diagram:
ρ
f ∗E
-
f ∗ (η)
E
η
?
?
Y
f
-
X
For y ∈ Y , f ∗ Ey = f ∗ (η)−1 (y) = η −1 f (y) = Ef (y) and hence f ∗ E is a vector bundle over Y .
The Whitney sum E ⊕ F of two vector bundles η : E → X and φ : F → X is the
family of all vector spaces Ex ⊕ Fx , given the subspace topology from E × F . Since E and
F are locally trivial and satisfy the compatibility condition, the same is true for E ⊕ F , so
E ⊕ F is a vector bundle.
We provide some familiar examples of vector bundles on smooth manifolds:
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• The manifold X × Rn together with the projection map onto the first coordinate is
called the trivial vector bundle of rank n over X.
• If X = S 1 , then the trivial line bundle over S 1 is an infinite cylinder.
• Consider the real projective space RPn . Recall that we can view this in two different
ways: either as S n with antipodal points identified, or as the set of lines in Rn+1 passing through the origin. We may define the tautological bundle τ n by setting the
fiber τxn to be the line corresponding to x in the second description. Then we may set
τ n := qx∈X τxn .
Is τ n trivial?
To answer this question, consider the case n = 1, and we turn to the first description of the projective plane. Drawing the half-circle, and the oriented fibers, we can
see that we stil have local triviality, so τ 1 is in fact a vector bundle, but the gluing
reverses the orientation on the boundary line, so τ 1 is an infinite Mobius band, and
therefore not trivial.
• We define the tangent bundle over X by T X := qx∈X Tx X, where Tx X is the tangent
space of X at x: the set of all tangent vectors at x.
• If X ⊂ Rm , the we define the normal bundle N X = qx∈X Nx X, where Nx X is a
linear subspace of Rm orthogonal to Tx X. If we consider S n ⊂ Rn+1 , then N S n is the
trivial line bundle.
A smooth manifold X is called parallelizable if its tangent bundle T X is trivial.
• S 1 is parallelizable, by taking the tangent vectors to be unit vectors pointing in the
clockwise direction, we have T S 1 ∼
= S 1 × R.
• S 2 is not parallelizable, by the Hairy Ball Theorem: There is no nonvanishing
continuous tangent vector field on S 2n . This was proposed by Poincaré in the 19th
century and first proved by Brouwer in 1912.
• The torus S 1 × S 1 is parallelizable: Recall that we can construct a torus from a piece
of paper by gluing together the opposite sides of the paper (without twisting). If we
were to do this with graph paper, then the grid lines give an idea as to what these
tangent vectors would look like. So, one may say that it is possible to comb a hairy
doughnut.
• S n is parallelizable only for n = 1, 3, 7.
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Stiefel-Whitney Classes
Let H i (X, G) be the i-th singular cohomology group of X with coefficients in G. P
For a vector
bundle E over X, we define the total Stiefel-Whitney class of E by w(E) = i≥0 wi (E),
where wi (E) ∈ H i (X, Z/2). We may then define the wi (E) axiomatically:
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• (rank) w0 (E) = 1 for any vector bundle E over X. Furthermore, wk (E) = 0 for all
k > rank(E).
• (naturality) If f : Y → X is a continuous map, and f ∗ (E) is the pullback bundle, then
wi (f ∗ (E)) = f ∗ (wi (E)).
P
• (Whitney sum) For vector bundles E, F over X, wk (E ⊕ F ) = i+j=k wi (E)wj (F ).
Using the total Stiefel-Whitney class, we may write this as w(E ⊕ F ) = w(E)w(F ).
• (normalization) Let τ be the tautological line bundle over RP1 , then w1 (τ ) 6= 0.
The fourth property assures that these classes are not all trivial. Existence of the StiefelWhitney classes can be shown through various contructions, and for a proof of uniqueness,
we may refer to Milnor-Stasheff.
A corresponding characteristic class can be defined
P if we change the base field to C. We
define the total Chern class of E by c(E) = i≥0 ci (E), where ci (E) ∈ H 2i (X, Z). This
can again be defined axiomatically: the first 3 axioms will remain the same, but the normalization condition is a little more complicated (to me). We take c(τ k ) = 1 − H, where H is
Poincaré dual to the syperplace CPk−1 ⊂ CPk .
We may determine certain properties of these classes by means of the above axioms:
• (triviality) We note that by the second axiom, we have w(n ) = 1 for any smooth
manifold X. This is due to the fact that if E is a trivial bundle over X, then there
exists a homomorphism from E to a vector bundle over a point. The i-th cohomology
groups of a point are trivial for all i ≥ 1, so wi (E) = 0 for all i ≥ 1.Hence, these
characteristic classes can be seen to measure the “triviality” of a vector bundle.
• (well-defined) Again by the second axiom, we see that if E ∼
= F , then w(E) = w(F ),
but the converse does not hold in general.
• (stability) Combining this with the Whitney sum formula, it can be seen that for any
vector bundle E over X, w(E ⊕ n ) = w(E)w(n ) = w(E).
• (classification of line bundles) Since a line bundle L has rank 1, w(L) = 1 + w1 (L). It
turns out that these classes are enough to completely classify line bundles. In fact, we
have a group isomorphism
w1 : V ect1 (X) → H 1 (X, Z/2) given by [L1 ⊗ L2 ] 7→ w1 (L1 ) + w1 (L2 ).
For example, H 1 (RP1 , Z/2) ∼
= Z/2, and so all line bundles over RP1 are isomorphic either to the trivial bundle or the tautological line bundle. On the other hand,
H 1 (S 1 , Z/2) ∼
= Z, and so each line bundle over S 1 is sent to the number of “twists’.
However, the Stiefel-Whitney classes are not enough to completely classify vector bundles
of higher rank. We consider for example the n-sphere S n ⊂ Rn+1 . For every point x ∈ S n ,
the fiber Nx S n of the normal bundle over S n is the line passing though the origin and the
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point x. Thus, N S n is a trivial bundle, and so w(N S n ) = 1. Furthermore, we can see that
N Sn ⊕ T Sn ∼
= S n × Rn+1 , so by the Whitney sum formula, w(T S n ) = w(T S n ⊕ N S n ) = 1.
But for instance, we know from the Hairy Ball Theorem that T S 2 is not trivial. We say that
T S 2 is “stably trivial”.
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Dimension of a real division algebra
Consider the real projective space RPn . The group H i (RPn , Z/2) is cyclic of order 2 for
each 0 ≤ i ≤ n, and is trivial for i > n. So, we may view H ∗ (RPn , Z/2) as the Z/2-algebra
Z/2[t]/(tn+1 ). Then, the Stiefel-Whitney class of a vecotr bundle E over RPn will be a
polynomial w(E) = 1 + w1 (E)t + ... · · · + wn (E)tn . In particular, by the 4th axiom, we have
w(τ ) = 1 + t.
Theorem 3.1. Let τ be the tautological line bundle over RPn and let 1 be the trival line
bundle. Then T RPn ⊕ 1 is isomorphic to the (n + 1)-fold Whitney sum τ ⊕ · · · ⊕ τ . Hence,
w(T RPn ) = w(T RPn ⊕ 1 ) = w(τ ⊕ · · · ⊕ τ ),
and by the Whitney sum formula,
w(τ ⊕ · · · ⊕ τ ) = w(τ )n+1 = (1 + t)n+1 .
Corollary 3.2. If RPn is parallelizable, then n + 1 = 2r for some integer r ≥ 0.
Proof. If RPn is parallelizable, then by defintion, T RPn is trivial, and so w(T RPn ) = 1. We
have w(T RPn ) = (1 + t)n+1 and so,
(1 + t)n+1 = 1 ∈ Z/2[t]/(tn+1 )
⇒ (1 + t)n+1 = 1 + tn+1 ∈ Z/2[t]
⇒ n + 1 = 2r .
r
Warning: This does not say that RP2 −1 is parallelizable for all r ≥ 0, only that these are
the only real projective spaces which have a chance at being parallelizable. In fact, it can
r
be shown that only RP1 , RP3 , andRP7 are actually parallelizable, and that RP2 −1 is not
parallelizable for r > 3.
We may now provide an application to finite-dimensional real division algebras.
Let V be a real division algebra of finite dimension n. That is, V is an n-dimensional
vector space over R together with a multiplication map V × V → V such that if x, y ∈ V
and both are non-zero, then x · y 6= 0 (ie. V has no zero-divisors). This map is an R-linear,
associative map, and is equivalent to the smooth map V ⊗R V → V . Thus, we can make
RP(V ) into a Lie group using the division algebra structure. (This makes sense since the
product of lines through the origin is again a line through the origin). We take as a fact that
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all Lie groups are parallelizable, and thus, RP(V ) is parallelizable.
Now, since dim(V ) = n, RP(V ) ∼
= RPn−1 and so by our above theorem, we must have
r
n = 2 for some r ≥ 0.
Theorem 3.3. If V is a finite-dimensional real division algebra, then dim(V ) = 2r for some
integer r ≥ 0.
Corollary 3.4. Up to isomorphism, the only finite-dimensional real division algebras are:
the real numbers, the complex numbers, the quaternions and the Cayley numbers.
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