Degree

Now that the basic properties of homology have been established, we can begin to move a little more freely. Our first topic, exploiting the calculation of H_n(Sⁿ) , is Brouwer’s notion of degree for mapsSⁿ

→

^Sn. Historically, Brouwer’s introduction of this concept in the years 1910–12 preceded the rigorous development of homology, so his definition was rather different, using the technique of simplicial approximation which we explain in§2.C. The later definition in terms of homology is certainly more elegant, though perhaps with some loss of geometric intuition. More in the spirit of Brouwer’s definition is a third approach using differential topology, presented very lucidly in [Milnor 1965].

the antipodal map. Note that the antipodal map has no fixed points, so the fact that maps without fixed points are homotopic to the antipodal map is a sort of converse statement.

Here is an interesting application of degree:

T

heorem 2.28. Sⁿ has a continuous field of nonzero tangent vectors iff n is odd.

P

roof: Suppose x

,

v(x) is a tangent vector field on Sⁿ, assigning to a vector x∈ Sⁿ the vectorv(x) tangent to Sⁿ atx . Regarding v(x) as a vector at the origin instead of at x , tangency just means that x and v(x) are orthogonal in Rⁿ⁺¹. If v(x)≠ 0 for all x , we may normalize so that |v(x)| = 1 for all x by replacing v(x) byv(x)/|v(x)|. Assuming this has been done, the vectors (cos t)x + (sin t)v(x) lie in the unit circle in the plane spanned by x and v(x) . Letting t go from 0 to π , we obtain a homotopyf_t(x)= (cos t)x + (sin t)v(x) from the identity map of Sⁿ to the antipodal map −¹1 . This implies that deg(−¹1)= deg¹1 , hence (−1)ⁿ⁺¹ = 1 and n must be odd.

Conversely, ifn is odd, say n= 2k − 1, we can define v(x1, x₂,··· , x_2k−1, x_2k)= (−x₂, x₁,··· , −x_2k, x_2k−1) . Then v(x) is orthogonal to x , so v is a tangent vector

field onSⁿ, and |v(x)| = 1 for all x ∈ Sⁿ. tu

For the much more difficult problem of finding the maximum number of tan-gent vector fields on Sⁿ that are linearly independent at each point, see [VBKT] or [Husemoller 1966].

Another nice application of degree, giving a partial answer to a question raised in Example 1.43, is the following result:

P

roposition 2.29. Z2 is the only nontrivial group that can act freely on Sⁿ if n is even.

Recall that an action of a group G on a space X is a homomorphism from G to the group Homeo(X) of homeomorphisms X

→

X , and the action is free if the homeomorphism corresponding to each nontrivial element of G has no fixed points.

In the case of Sⁿ, the antipodal mapx

,

−x generates a free action of Z₂.

P

roof: Since the degree of a homeomorphism must be ±1, an action of a group G on Sⁿ determines a degree function d : G

→

{±1}. This is a homomorphism since degf g = deg f deg g . If the action is free, then d sends every nontrivial element of G to (−1)ⁿ⁺¹ by property (g) above. Thus when n is even, d has trivial kernel, so

G⊂ Z₂. tu

We shall next describe a technique for computing degrees which can be applied to most maps that arise in practice. Suppose f : Sⁿ

→

^Sn has the property that for

some point y ∈ Sⁿ, the preimage f⁻¹(y) consists of only finitely many points, say x₁,··· , x_m. LetU₁,··· , U_m be disjoint neighborhoods of these points, mapped byf into a neighborhood V of y . Then f (U_i− x_i) ⊂ V − y for each i, and we have a commutative diagram

−−−−−→ −−−−−→

−−−−−→ −−−−−→

−−−−−→

− − −−−→

^{H (n S )}

f j

k

∗

f∗

−−−−−→

^f∗

nSⁿ y

≈

≈

−−−−−−−→

− −− − −−

− −→

− −−−−

− −−→

H (_n Sⁿ) H (_n Sⁿ)

≈

≈ ,

-n S

H ( ⁿ,Sⁿ

-

x )

n V

H ( ,V

-

y)

n i

i

pi

i

i

Ui

H ( ,U

-

x)

n S

H

(

ⁿ,Sⁿ

-

f^-1( )y

)

where all the maps are the obvious ones, in particulark_i and p_i are induced by inclu-sions. The two isomorphisms in the upper half of the diagram come from excision, while the lower two isomorphisms come from exact sequences of pairs. Via these four isomorphisms, the top two groups in the diagram can be identified with H_n(Sⁿ)≈ Z, and the top homomorphism f_∗ becomes multiplication by an integer called thelocal degree of f at x_i, written degf ||xi.

For example, if f is a homeomorphism, then y can be any point and there is only one corresponding x_i, so all the maps in the diagram are isomorphisms and degf ||xi= deg f = ±1. More generally, if f maps each Ui homeomorphically onto V , then deg f ||xi= ±1 for each i. This situation occurs quite often in applications, and it is usually not hard to determine the correct signs.

Here is the formula that reduces degree calculations to computing local degrees:

P

roposition 2.30. deg f =P

idegf ||xi.

P

roof: By excision, the central term H_n Sⁿ, Sⁿ− f⁻¹(y)

in the preceding diagram is the direct sum of the groups H_n(U_i, U_i− xi) ≈ Z, with ki the inclusion of the i^thsummand. Since the upper triangle commutes, the projections of this direct sum onto its summands are given by the maps p_i. Identifying the outer groups in the diagram with Z as before, commutativity of the lower triangle says that p_ij(1)= 1, hence j(1)= (1, ··· , 1) =P

ik_i(1) . Commutativity of the upper square says that the middle f_∗ takes k_i(1) to deg f ||xi, hence P

ik_i(1)= j(1) is taken to P

idegf ||xi. Commutativity of the lower square then gives the formula degf =P

idegf ||xi. tu

E

xample 2.31. We can use this result to construct a map Sⁿ

→

^Snof any given degree, for each n≥ 1. Let q : Sⁿ

→

^WkSⁿ be the quotient map obtained by collapsing the complement ofk disjoint open balls B_iinSⁿto a point, and letp :W

kSⁿ

→

^Snidentify

all the summands to a single sphere. Consider the compositionf = pq. For almost all y ∈ Sⁿ we have f⁻¹(y) consisting of one point x_i in eachB_i. The local degree off atx_i is±1 since f is a homeomorphism near x_i. By precomposingp with reflections of the summands ofW

kSⁿ if necessary, we can make each local degree either +1 or

−1, whichever we wish. Thus we can produce a map Sⁿ

→

^{Sn of degree ±k.}

E

xample 2.32. In the case of S¹, the map f (z)= z^k, where we view S¹ as the unit circle in C, has degree k. This is evident in the case k = 0 since f is then constant.

The case k < 0 reduces to the case k > 0 by composing with z

,

^z−1, which is a reflection, of degree −1. To compute the degree when k > 0, observe first that for any y ∈ S¹,f⁻¹(y) consists of k points x₁,··· , xk near each of which f is a local homeomorphism. Near each x_i the map f can be homotoped, stretching by a factor of k without changing local degree, to become the restriction of a rotation of S¹. A rotation has degree +1 since it is homotopic to the identity, and since a rotation is a homeomorphism, its degree equals its local degree at any point. Hence degf ||xi= 1 and degf = k.

Another way of obtaining a map Sⁿ

→

^{Sn of degree} k is to take a repeated sus-pension of the map z

,

^zk in Example 2.32, since suspension preserves degree:

P

roposition 2.33. deg Sf = deg f , where Sf : Sⁿ⁺¹

→

^Sn+1 is the suspension of the map f : Sⁿ

→

^Sn.

P

roof: Let CSⁿ denote the cone (Sⁿ×I)/(Sⁿ×1) with base Sⁿ = Sⁿ×0 ⊂ CSⁿ, so CSⁿ/Sⁿ is the suspension of Sⁿ. The map f induces Cf : (CSⁿ, Sⁿ)

→

^{(CSn, Sn)}

with quotientSf . The naturality of the boundary maps

−−−−−→

Sf

−−−−−→ −−−−−→

H ( ) H (_n S )

∗ f_∗

∼

∼n 1+ S^{n 1+ n}

≈∂

−−−−−→

H∼ ( ) H (∼_n S ) S^{n 1+ n}

n 1+

≈∂ in the long exact sequence of the pair (CSⁿ, Sⁿ) then

gives commutativity of the diagram at the right. Hence iff_∗ is multiplication by d , so is Sf_∗. tu

Note that forf : Sⁿ

→

^Sn, the suspension Sf maps only one point to each of the two ‘poles’ ofSⁿ⁺¹. This implies that the local degree of Sf at each pole must equal the global degree of Sf . Thus the local degree of a map Sⁿ

→

^Sn can be any integer ifn≥ 2, just as the degree itself can be any integer when n ≥ 1.

在文檔中 Allen Hatcher (頁 143-146)