Elementary Applications

us-ing the natural identification (X×Y )/(X×B ∪ A×Y ) = X/A ∧ Y /B . This works when A= ∅ since we interpret X/∅ as X₊, and similarly if Y = ∅. Via the diagonal map we obtain also a product Kⁱ(X, A)⊗K^j(X, B)

→

^{Ki+j(X, A∪ B).}

With these definitions the preceding two propositions are valid also for unreduced K–groups.

2.3. Division Algebras and Parallelizable Spheres

With the hard work of proving Bott Periodicity now behind us, the goal of this section is to prove Adams’ theorem on the Hopf invariant, with its famous applications including the nonexistence of division algebras beyond the Cayley octonions:

T

heorem 2.16. The following statements are true only for n= 1, 2, 4, and 8:

(a) Rⁿ is a division algebra.

(b) Sⁿ⁻¹ is parallelizable, i.e., there exist n− 1 tangent vector fields to Sⁿ⁻¹ which are linearly independent at each point, or in other words, the tangent bundle to Sⁿ⁻¹ is trivial.

A division algebra structure on Rⁿ is a multiplication map Rⁿ×Rⁿ

→

^{Rn such}

that the maps x

,

^{ax and x}

,

xa are linear for each a∈ Rⁿand invertible if a≠ 0.

Since we are dealing with linear maps Rⁿ

→

^Rn, invertibility is equivalent to having trivial kernel, which translates into the statement that the multiplication has no zero divisors. An identity element is not assumed, but the multiplication can be modified to produce an identity in the following way. Choose a unit vector e ∈ Rⁿ. After composing the multiplication with an invertible linear map Rⁿ

→

^{Rn taking e2 to e}

we may assume that e²= e. Let α be the map x

,

xe and β the map x

,

ex . The new product (x, y)

,

^{α−1(x)β−1}(y) then sends (x, e) to α⁻¹(x)β⁻¹(e)= α⁻¹(x)e= x , and similarly it sends (e, y) to y . Since the maps x

,

^{ax and x}

,

xa are surjective for each a≠ 0, the equations ax = e and xa = e are solvable, so nonzero elements of the division algebra have multiplicative inverses on the left and right.

H-Spaces

The first step in the proof of the theorem is to reduce it to showing when the sphere Sⁿ⁻¹ is an H–space.

To say that Sⁿ⁻¹ is an H–space means there is a continuous multiplication map Sⁿ⁻¹×Sⁿ⁻¹

→

^Sn−1 having a two-sided identity element e∈ Sⁿ⁻¹. This is weaker than being a topological group since associativity and inverses are not assumed. For exam-ple, S¹, S³, and S⁷are H–spaces by restricting the multiplication of complex numbers, quaternions, and Cayley octonions to the respective unit spheres, but only S¹ and S³ are topological groups since the multiplication of octonions is nonassociative.

L

emma 2.17. If Rⁿ is a division algebra, or if Sⁿ⁻¹ is parallelizable, then Sⁿ⁻¹ is an H–space.

P

roof: Having a division algebra structure onRⁿ with two-sided identity, an H–space structure on Sⁿ⁻¹is given by (x, y)

,

xy/|xy|, which is well-defined since a division algebra has no zero divisors.

Now suppose that Sⁿ⁻¹ is parallelizable, with tangent vector fields v₁,··· , v_n−1 which are linearly independent at each point of Sⁿ⁻¹. By the Gram-Schmidt process we

may make the vectors x, v₁(x),··· , v_n−1(x) orthonormal for all x∈ Sⁿ⁻¹. We may assume also that at the first standard basis vector e₁, the vectors v₁(e₁),··· , v_n−1(e₁) are the standard basis vectors e₂,··· , e_n, by changing the sign of v_n−1 if necessary to get orientations right, then deforming the vector fields near e₁. Let α_x ∈ SO(n) send the standard basis to x, v₁(x),··· , v_n−1(x) . Then the map (x, y)

,

^αx(y) defines an H–space structure on Sⁿ⁻¹ with identity element the vector e₁ since α_e1 is the

identity map and α_x(e₁)= x for all x . tu

Before proceeding further let us list a few easy consequences of Bott periodicity which will be needed.

(1) We have already seen that eK(Sⁿ) is Z for n even and 0 for n odd. This comes from repeated application of the periodicity isomorphism eK(X)≈ eK(S²X) , α

,

α∗ (H − 1), the external product with the generator H − 1 of eK(S²) , where H is the canonical line bundle over S²= CP¹. In particular we see that a generator of K(Se ^2k) is the k fold external product (H− 1) ∗ ··· ∗ (H − 1). We note also that the multiplication in eK(S^2k) is trivial since this ring is the k fold tensor product of the ring eK(S²) , which has trivial multiplication by Corollary 2.3.

(2) The external product eK(S^2k)⊗ eK(X)

→

^{K(Se 2k}∧ X) is an isomorphism since it is an iterate of the periodicity isomorphism.

(3) The external product K(S^2k)⊗K(X)

→

^K(S2k×X) is an isomorphism. This fol-lows from (2) by the same reasoning which showed the equivalence of the reduced and unreduced forms of Bott periodicity. Since external product is a ring homo-morphism, the isomorphism eK(S^2k∧ X) ≈ eK(S^2k)⊗ eK(X) is a ring isomorphism.

For example, since K(S^2k) can be described as the quotient ring Z[α]/(α²) , we can deduce that K(S^2k×S^{2`</sup>) isZ[α, β]/(α<sup>2</sup>, β<sup>2</sup>) where α and β are the pullbacks of generators of eK(S<sup>2k</sup>) and eK(S<sup>2`}) under the projections of S^2k×S^{2`</sup> onto its two factors. An additive basis for K(S<sup>2k</sup>×S<sup>2`}) is thus {1, α, β, αβ}.

We can apply the last calculation to show that S^2k is not an H–space if k > 0 . Suppose µ : S^2k×S^2k

→

^S2k is an H–space multiplication. The induced homomor-phism of K–rings then has the form µ^∗:Z[γ]/(γ²)

→

^{Z[α, β]/(α2, β2}) . We claim that µ^∗(γ)= α + β + mαβ for some integer m. The composition S^2k

---→

^{i S2k×S2k}

---→

^{µ S2k}

is the identity, where i is the inclusion onto either of the subspaces S^2k×{e} or {e}×S^2k, with e the identity element of the H–space structure. The map i^∗ for i the inclusion onto the first factor sends α to γ and β to 0 , so the coefficient of α in µ^∗(γ) must be 1 . Similarly the coefficient of β must be 1 , proving the claim. However, this leads to a contradiction since it implies that µ^∗(γ²) = (α + β + mαβ)² = 2αβ ≠ 0, which is impossible since γ²= 0.

There remains the much more difficult problem of showing that Sⁿ⁻¹ is not an H–space when n is even and different from 2 , 4 , and 8 . The first step is a simple construction which associates to a map g : Sⁿ⁻¹×Sⁿ⁻¹

→

^{Sn−1 a map g : Sb 2n−1}

→

^Sn.

To define this, we regard S²ⁿ⁻¹ as ∂(Dⁿ×Dⁿ) = ∂Dⁿ×Dⁿ∪ Dⁿ×∂Dⁿ, and Sⁿ we take as the union of two disks Dⁿ₊ and D₋ⁿ with their boundaries identified. Then

b

g is defined on ∂Dⁿ×Dⁿ by g(x, y)b = |y|g(x, y/|y|) ∈ Dⁿ₊ and on Dⁿ×∂Dⁿ by b

g(x, y) = |x|g(x/|x|, y) ∈ D₋ⁿ. Note that g is well-defined and continuous, evenb when|x| or |y| is zero, and bg agrees with g on Sⁿ⁻¹×Sⁿ⁻¹.

Now we specialize to the case that n is even, or in other words, we replace n by 2n . For a map f : S⁴ⁿ⁻¹

→

^{S2n, let Cf be S2n} with a cell e⁴ⁿ attached by f . The quotient C_f/S²ⁿ is then S⁴ⁿ, and since eK¹(S⁴ⁿ)= eK¹(S²ⁿ)= 0, the exact sequence of the pair (C_f, S²ⁿ) becomes a short exact sequence

0

-→

^{K(Se 4n)}

-→

^{K(Ce f)}

-→

^{K(Se 2n)}

-→

⁰

Let α∈ eK(C_f) be the image of the generator (H− 1) ∗ ··· ∗ (H − 1) of eK(S⁴ⁿ) and let β∈ eK(C_f) map to the generator (H− 1) ∗ ··· ∗ (H − 1) of eK(S²ⁿ) . The element β² maps to 0 in eK(S²ⁿ) since the square of any element of eK(S²ⁿ) is zero. So by exactness we have β²= hα for some integer h. The mod 2 value of h depends only on f , not on the choice of β , since β is unique up to adding an integer multiple of α , and (β+ mα)² = β²+ 2mαβ since α² = 0. The value of h mod 2 is called the mod 2 Hopf invariant of f . In fact αβ= 0 so h is well-defined in Z not just Z2, as we will see in§4.1, but for our present purposes the mod 2 value of h suffices.

L

emma 2.18. If g : S²ⁿ⁻¹×S²ⁿ⁻¹

→

^S2n−1 is an H–space multiplication, then the as-sociated map g : Sb ⁴ⁿ⁻¹

→

^S2n has Hopf invariant±1.

P

roof: Let e∈ S²ⁿ⁻¹ be the identity element for the H–space multiplication, and let f = bg . In view of the definition of f it is natural to view the characteristic map Φ of the 4n cell of C_f as a map (D²ⁿ×D²ⁿ, ∂(D²ⁿ×D²ⁿ))

→

^{(Cf, S2n}) . In the following commutative diagram the horizontal maps are the product maps. The diagonal map is external product, equivalent to the external product eK(S²ⁿ)⊗ eK(S²ⁿ)

→

^{K(Se 4n) , which}

is an isomorphism since it is an iterate of the Bott periodicity isomorphism.

D e

K∼( ²ⁿ×{ },∂D²ⁿ×{e}) K∼({e}×D²ⁿ,{e}×∂D²ⁿ)

K∼(D²ⁿ×D²ⁿ,∂D²ⁿ×D²ⁿ) K∼(D²ⁿ×D²ⁿ,D²ⁿ× ∂D²ⁿ)

−−−−−→

K∼

(

D²ⁿ×D²ⁿ,∂(D²ⁿ×D²ⁿ)

)

K∼(C ,D²ⁿ)

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→

−−−−−−−−−−−−−−−−−−−−−−− −−−−−−−→

f - K∼(C ,D²ⁿ)

f K∼(C ,S²ⁿ)

+ f

K C∼( ^f) ^{K C∼( f)}

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→

^{K C∼( f)}

−−→ −−→ −−→

≈

≈ ≈

−−→ −−→

_≈

⊗

⊗

⊗

⊗

⊗

Φ^∗ Φ^∗

Φ^∗

By the definition of an H–space and the definition of f , the map Φ restricts to a homeomorphism from D²ⁿ×{e} onto D²ⁿ₊ and from{e}×D²ⁿ onto D²ⁿ₋ . It follows that the element β⊗β in the upper left group maps to a generator of the group in the bottom row of the diagram, since β restricts to a generator of eK(S²ⁿ) by definition.

Therefore by commutativity of the diagram, the product map in the top row sends β⊗β to±α since α was defined to be the image of a generator of eK(C_f, S²ⁿ) . Thus we have β²= ±α, which says that the Hopf invariant of f is ±1. tu

In view of this lemma, Theorem 2.16 becomes a consequence of the following theorem of Adams:

T

heorem 2.19. If f : S⁴ⁿ⁻¹

→

^S2n is a map whose mod 2 Hopf invariant is 1 , then n= 1, 2, or 4.

The proof of this will occupy the rest of this section.

在文檔中 1.2. Classifying Vector Bundles (頁 62-66)