1. Chain complexes Definition. A sequence of abelian groups ...C−2, C−1, C0, C1,... with ...

21 Pages·2015·0.28 MB·English

1. Chain complexes
Definition. A sequence of abelian groups
...C , C , C , C ,...
−2 −1 0 1
with homomorphisms ∂ : C → C is called a (homological) chain complex if
i i+1 i
∂ ◦∂ =0 for all i∈Z.
i−1 i
A cohomological chain complex is almost the same thing, but with reversed
grading: a sequence of abelian groups
...C−2, C−1, C0, C1,...
together with homomorphisms di: Ci−1 →Ci such that di◦di−1 =0 for all i∈Z.
Wewillconcentrateonhomologicalchaincomplexes;allresultsholdanalogously
for cohomological chain complexes.
A chain complex (or just a sequence of abelian groups with homomorphisms) is
called bounded below (bounded above) if C = 0 for i (cid:28) 0 (resp. i (cid:29) 0). It
i
is called non-negatively graded (non-positively graded) if C = 0 for i < 0
i
(resp. i>0).
Definition. We call the subgroup Z (C ) = ker(∂ : C → C ) < C the sub-
i • i−1 i i−1 i
group of i-cycles and the subgroup B (C ) = im(∂ : C → C ) < C the sub-
i • i i+1 i i
group of i-boundaries.
Lemma 1.1. For any chain complex C , B (C ) is a subgroup of Z (C ).
• i • i •
Definition. The ith homology group of a chain complex C is defined as the
•
quotient group
H (C )=Z (C )/B (C ).
i • i • i •
If Z = B for all n (and thus H = 0), we call C exact or acyclic. An exact
n n n •
chain complex is more usually called exact sequence. An exact sequence of the
form
0→A(cid:48) →A→A(cid:48)(cid:48) →0
is often called a short exact sequence.
Lemma 1.2. Let A, B, C be abelian groups and f: A → B and g: B → C
homomorphisms.
f
(1) 0→A−→B is exact iff f is injective.
f
(2) A−→B →0 is exact iff f is surjective.
f
(3) 0→A−→B →0 is exact iff f is an isomorphism.
f g
(4) 0 → A −→ B −→ C → 0 is exact iff f is injective, g is surjective, and
kerg =imf.
Definition. Let A , B be sequences of abelian groups and homomorphisms (or
• •
chain complexes). A map of sequences (or map of chain complexes) is a
commutative diagram
··· A ∂nA A ∂nA−1 A ···
n+1 n n−1
fn+1 fn fn−1
··· B ∂nB B ∂nB−1 B ···
n+1 n n−1
1
2
Lemma 1.3. A map of chain complexes f: C →D induces maps
• •
Z(f): Z (C )→Z (D ) of n-cycles,
n • n •
B(f): B (C )→B (D ) of n-boundaries, and
n • n •
H (f)=f : H (C )→H (D ) on homology.
n ∗ n • n •
Lemma 1.4 (Five-lemma). Let
A A A A A
1 2 3 4 5
f1 f2 f3 f4 f5
B B B B B
1 2 3 4 5
be a commutative diagram of abelian groups with exact rows. Then:
(1) if f , f are surjective and f is injective then f is surjective.
2 4 5 3
(2) if f , f are injective and f is surjective then f is injective.
2 4 1 3
(3) in particular, if f , f , f , f are isomorphisms then so is f .
1 2 4 5 3
Definition. A short exact sequence of the form 0 → A(cid:48) → A(cid:48)⊕A(cid:48)(cid:48) → A(cid:48)(cid:48), where
the first map is the inclusion into the first summand and the second map is the
projection onto the second, is called split exact.
See homework problem 1.2 for characterizations of split exact sequences.
Definition. Let f: A → B be a homomorphism between abelian groups. Define
its cokernel coker(f) to be the quotient group B/im(f) and its coimage coim(f)
to be A/ker(f).
Lemma 1.5. For any homomorphism f: A→B of abelian groups, we have:
(1) f: coim(f)→im(f) is an isomorphism;
f
(2) 0→ker(f)→A−→B →coker(f)→0 is exact.
Lemma 1.6 (Snake lemma). Given a diagram of abelian groups
A A A 0
1 2 3
0 B B B
1 2 3
(1.7)
with exact rows. Let K denote the kernel of A → B and C its cokernel. Then
i i i i
there is a “snake homomorphism” K →C such that the sequence
3 1
K →K →K →C →C →C
1 2 3 1 2 3
3
is exact:
0 0 0
K K K
1 2 3
A A A 0
1 2 3
0 B B B
1 2 3
C C C .
1 2 3
0 0 0
If A →A is injective then so is K →K , and if B →B is injective then so is
1 2 1 2 2 3
C →C .
2 3
Furthermore,thesnakemapis natural,meaningthatifwehaveamap(A ,B )→
i i
(A(cid:48),B(cid:48)) of diagrams of the type (1.7) then the following square commutes:
i i
K C
3 1
K(cid:48) C(cid:48).
3 1
i p
Theorem 1.8. Let 0 → A →− B −→ C → 0 be a short exact sequence of chain
• • •
complexes (meaning 0 → A → B → C → 0 is exact for each n ∈ Z). Then
n n n
there is a connecting homomorphism δ : H (C ) → H (A ) such that the
n n+1 • n •
following long sequence is exact:
···−p→∗ H (C )−δ→n H (A )−i→∗ H (B )−p→∗ H (C )−δ−n−−→1 H (A )−i→∗ ··· .
n+1 • n • n • n • n−1 •
The homomorphism δ is natural: given a map of short exact sequences of chain
complexes (A ,B ,C )→(A(cid:48),B(cid:48),C(cid:48)), the following square commutes:
• • • • • •
δ
H (C ) H (A )
n+1 • n •
H (C(cid:48)) δ H (A(cid:48)).
n+1 • n •
2. Categories and functors
Definition. A category C consists of:
• a class ob(C) of objects;
• for each pair of objects X, Y ∈ob(C), a set of morphisms Hom (X,Y);
C
• for each object X ∈ob(C), an element id ∈Hom (X,X) called identity
X C
morphism;
4
• for each three objects X, Y, Z ∈ob(C), a map
◦: Hom (Y,Z)×Hom (X,Y)→Hom(X,Z), (g,f)(cid:55)→g◦f
C C
called composition.
These have to satisfy the following axioms:
(1) The composition ◦ is associative;
(2) For f ∈Hom (X,Y), id ◦f =f and f ◦id =f.
C Y X
A morphism f ∈ Hom (X,Y) is called an isomorphism (and the objects X, Y
C
isomorphic) if there is another morphism g ∈Hom (Y,X) such that g◦f =id
C X
and g◦g =id . If such a g exists, it is unique and is denoted by f−1.
Y
We will often abuse notation and write X ∈ C for X ∈ ob(C), f ∈ Hom(X,Y)
or even just f: X → Y for f ∈ Hom (X,Y), and id for id . We will also use
C X
commutative diagrams to denote equalities between compositions of morphisms.
Definition. We use the following standard notations for familiar categories:
Set: The category of sets and functions;
Ab: The category of abelian groups and homomorphisms;
Top: The category of topological spaces and continuous maps.
Definition. Let C, D be categories. A (covariant) functor F: C → D consists
of:
• a function ob(C)→ob(D), also called F; and
• for every X, Y ∈ ob(C), a function Hom (X,Y) → Hom (F(X),F(Y))
C D
denoted by f (cid:55)→F(f) or f (cid:55)→f
∗
satisfying (id ) =id and (g◦f) =g ◦f .
X ∗ F(X) ∗ ∗ ∗
A contravariant functor F: C →D consists of:
• a function ob(C)→ob(D), also called F; and
• for every X, Y ∈ ob(C), a function Hom (X,Y) → Hom (F(Y),F(X))
C D
denoted by f (cid:55)→F(f) or f (cid:55)→f∗
satisfying (id )∗ =id and (g◦f)∗ =f∗◦g∗. (“It turns arrows around.”)
X F(X)
Definition. A natural transformation η: F → G between two functors F,
G: C →D consistsofamorphismη ∈Hom (F(X),G(X))foreachobjectX ∈C
X D
such that for each morphism f: Hom (X,Y), the following diagram commutes:
C
F(X) ηX G(X)
F(f) G(f)
F(Y) ηY G(Y).
Natural transformations between contravariant functors are defined analogously.
A natural transformation η: F → G is called natural isomorphism (and F
and G isomorphic, F (cid:39)G) if η is an isomorphism for all X ∈C.
X
Definition. AcovariantfunctorF: C →Discalledanequivalenceofcategories
ifthereisanotherfunctorG: D →C suchthatG◦F (cid:39)Id andF◦G(cid:39)Id ,where
C D
Id , Id denote the identity functors on C and D, respectively.
C D
5
Definition. LetCbeacategoryand(X ) afamilyofobjectsinC,forsomeindex
i i∈I
set I. An object X together with morphisms ι : X → X is called coproduct of
i i
(cid:96)
the X , and is denoted by X , if for each test object Y ∈C, the map
i i∈I i
Hom (X,Y)−H−o−m−C−(ι−i,−−→) (cid:89)Hom (X ,Y)
C C i
i∈I
is a bijection. The coproduct of only two objects is denoted by X (cid:116)X .
1 2
Similarly, an object X with morphism π : X →X is called product of the X ,
i i i
(cid:81)
and is denoted by X , if for each test object Y ∈C, the map
i∈I i
Hom (Y,X)−H−o−m−C−(−−−,π−i→) (cid:89)Hom (Y,X )
C C i
i∈I
is a bijection. The product of only two objects is denoted by X ×X .
1 2
Lemma 2.1. In an arbitrary category C, (co-)products need not exist, but if they
do, they are unique up to isomorphism.
3. Rings and modules
Definition. A ring R is an abelian group together with a unity 1 ∈ R and an
associative bilinear map R×R → R, (x,y) (cid:55)→ xy, such that 1x = x1 = x for all
x∈R. A ring is called commutative if xy =yx for all x, y ∈R.
A map f: R → S between rings is called a ring homomorphism or map of
rings if it is linear, f(1 )=1 , and f(xy)=f(x)f(y) for all x, y ∈R.
R S
Definition. Aleft moduleM overaringR isanabeliangroupM togetherwith
a bilinear multiplication map R×M →M, (r,m)(cid:55)→r.m, such that 1.m=m and
(r r ).m=r .(r .m) for all m∈M, r ∈R.
1 2 1 2 i
A right module is an abelian group M with a bilinear multiplication map
M ×R → M, (m,r) (cid:55)→ m.r, such that m.1 = m and m.(r r ) = (m.r ).r for all
1 2 1 2
m∈M, r ∈R.
i
When we just say “module”, we agree to mean a left module.
Amapf: M →N betweentwo(leftorright)R-modulesM,N isanR-module
homomorphism if it is a abelian group homomorphism and f(r.m) = r.f(m)
(resp. f(m.r)=f(m).r) for all r ∈R, m∈M.
The category of left R-modules and R-module homomorphisms is denoted by
Mod .
R
(cid:81)
Definition. Theproductofafamily(M ) ofR-modules,denotedby M ,
i i∈I i∈I i
is the module whose underlying abelian group is the product groups, and the R-
module structure is given by r.((m ) ) = (r.m ) . The direct sum of the
i i∈I i i∈I
(cid:76)
family, denoted by M , is the submodule of families (m ) where all but
i∈I i i i∈I
finitely many m =0.
i
An R-module M is called free if it is isomorphic to an (arbitrarily indexed)
direct sum of copies of R.
Lemma 3.1. The direct product is a product in Mod in the category-theoretic
R
sense, and the direct sum is a coproduct.
Definition. Let R be a ring, M a right R-module, and N a left R-module. The
tensor product M ⊗ N is the abelian group obtained as follows. Denote by
R
Fr(M×N)thefreeabeliangroupwithgeneratorspairs(m,n)withm∈M,n∈N.
6
ThenM⊗ N isthequotientofFr(M×N)withrespecttoanequivalencerelation
R
∼ given by:
• (m +m ,n)∼(m ,n)+(m ,n)
1 2 1 2
• (m,n +n )∼(m,n )+(m,n )
1 2 1 2
• (m.r,n)∼(m,r.n)
We denote the equivalence class of (m,n) in M ⊗ N by m⊗n.
R
Proposition 3.2. In the context of the previous definition, let T be an abelian
group. Denote by Bil(M,N;T) the set of all bilinear homomorphisms f: M×N →
T with f(m.r,n)=f(m,r.n). Then there is a natural isomorphism
Bil(M,N;T)∼=Hom (M ⊗ N,T).
Z R
Definition (and lemma). An R-module M is called projective if it satisfies the
following equivalent conditions:
(1) For each diagram in Mod
R
M
N N 0
1 2
with exact row, a lift (dotted arrow) exists such that the resulting diagram
commutes.
(2) There is an R-module N such that M ⊕N is free.
(3) Every shot exact sequence 0→N →N →M →0 splits.
1 2
(4) The functor Hom (M,−) maps exact sequences to exact sequences (the
R
functor “is exact”).
Lemma3.3. Let0→N(cid:48) →N →N(cid:48)(cid:48) →0beanexactsequenceofrightR-modules,
and let M be a left R-module. Then the sequence of abelian groups
N(cid:48)⊗ M →N ⊗ M →N(cid:48)(cid:48)⊗ M →0
R R R
is exact. Let 0→N(cid:48) →N →N(cid:48)(cid:48) →0 be an exact sequence of left R-modules, and
let M be another left R-module. Then the sequence of abelian groups
0→Hom (N(cid:48)(cid:48),M)→Hom (N,M)→Hom (N(cid:48),M)
R R R
is exact.
Definition. A left R-module M is called flat if the functor −⊗ M from right R-
R
modulestoabeliangroupsisexact. ArightR-moduleisflatifthefunctorM⊗ −
R
from left R-modules to abelian groups is exact.
Lemma 3.4. Free modules are projective. Projective modules are flat. Not every
flat module is projective, and not every projective module is free.
4. Resolutions and derived functors
Definition. Let R be a ring. A nonnegatively graded chain complex P of R-
•
modulestogetherwithamap(cid:15)P →M (the“augmentation”)iscalledaprojective
0
resolution of M if
• For every i≥0, P is projective;
i
(cid:15)
• The extended chain complex ···→P →P →− M is exact.
1 0
7
Proposition 4.1. Every R-module M has a projective resolution.
Corollary 4.2. If R is a principal ideal domain then every R-module has a pro-
jective resolution of length 2:
0→P →P →M →0
1 0
Definition. Let C , D be nonnegatively graded chain complexes of R-modules
• •
and let f, g: C → D be two chain maps. A chain homotopy from f to g is a
• •
sequence of R-linear maps h : C →D such that
n n−1 n
g−f =h◦∂C +∂D◦h.
If such a chain homotopy exists, we call f and g chain homotopic and write
f (cid:39)g.
If f: C → D and g: D → C are chain maps with chain homotopies g◦f (cid:39)
• • • •
id and f ◦g (cid:39) id , we call f and g chain homotopy equivalences and the
C• D•
chain complexes C and D chain homotopy equivalent.
• •
Proposition 4.3. If f (cid:39)g then f =g : H (C )→H (D ).
∗ ∗ ∗ • ∗ •
Theorem 4.4. Let f: M → N be a morphism of R-modules, P → M a chain
•
complex where all P are projective, and N →N →0 be an exact complex. Then
i •
(1) Theexistsachainmapf : P →N makingthefollowingladdercommute:
• • •
··· P P M 0
1 0
f1 f0 f
··· N N N 0
1 0
(2) Any two such extensions f , g are chain homotopic.
• •
Corollary4.5. AnytwoprojectiveresolutionsofM arechainhomotopyequivalent.
Definition. Let R, S be two rings and F: Mod → Mod a (covariant or con-
R S
travariant) functor. We call F additive if the induced map on Hom-sets
F
Hom (M,N)−→Hom (F(M),F(N)) (resp. Hom (F(N),F(M)))
R S S
is a homomorphism of abelian groups.
Let F be an additive covariant functor as above. Then we call F
• left exact if 0→F(M(cid:48))→F(M)→F(M(cid:48)(cid:48)) is exact;
• right exact if F(M(cid:48))→F(M)→F(M(cid:48)(cid:48))→0 is exact;
• exactifitisrightandleftexact,i.e. if0→F(M(cid:48))→F(M)→F(M(cid:48)(cid:48))→
0 is exact
for all choices of exact sequences 0→M(cid:48) →M →M(cid:48)(cid:48) →0 of R-modules.
Similarly, if F is contravariant, we call it
• left exact if 0→F(M(cid:48)(cid:48))→F(M)→F(M(cid:48)) is exact;
• right exact if F(M(cid:48)(cid:48))→F(M)→F(M(cid:48))→0 is exact;
• exactifitisrightandleftexact,i.e. if0→F(M(cid:48)(cid:48))→F(M)→F(M(cid:48))→
0 is exact
for all choices of exact sequences 0→M(cid:48) →M →M(cid:48)(cid:48) →0 of R-modules.
8
Definition (and lemma). Let F: Mod →Mod be a covariant right exact func-
R S
tor, M an R-module, and P → M a projective resolution of M. Define the nth
•
left derived functor L F: Mod →Mod by
n R S
(L F)(N)=H (F(P )).
n n •
Similarly, if F is a contravariant left exact functor, define the n right derived
functor RnF: Mod →Mod by
R S
(RnF)(N)=Hn(F(P )).
•
Thisisindependentofthechoiceofresolutionandextendstoafunctorbydefining
it on morphisms as follows: if f: M →M(cid:48) is a morphism of R-modules, extend it
to a morphism f : P →P(cid:48) by Thm. 4.4 and set
• • •
L (F)(f)=H (F(f ));
n n •
similarly for right derived functors.
Lemma 4.6. If F is covariant right exact then L F = F. If F is contravariant
0
left exact then R0F =F.
Lemma 4.7. If R is a principal ideal ring and F: Mod → Mod a right exact
R S
covariant or left exact contravariant functor. Then L F = 0 (resp. RnF = 0) if
n
n≥2.
Lemma 4.8. Let F be a covariant left exact functor. Then L F =0 for all n≥1
n
if and only if F is exact.
Definition. Let R be a ring, M a right R-module, and N a left R-module. Define
TorR(M,N)tobethenthleftderivedfunctorofthefunctor−⊗ N: Mod→Ab,
n R R
applied to M:
TorR(M,N)=[L (−⊗ N)](M).
n n R
Let M and N be left modules. Define Extn(M,N) to be the nth right derived
R
functor of the functor Hom (−,N), applied to M:
R
Extn(M,N)=[RnHom(−,N)](M)
R
Proposition 4.9. (symmetry of Tor) The functor TorR coincides with the nth left
n
derived functor of the functor M ⊗ −: Mod →Ab, applied to N:
R R
TorR(M,N)=[L (M ⊗ −)](N).
n n R
5. Homology of spaces
Definition. DenotebyTopthecategoryoftopologicalspacesandcontinuousmaps.
WealsowriteTop forthecategoryof pointedspaces. Itsobjectsarepairs(X,x )
∗ 0
where X is a topological spaces and x ∈X. Morphisms from (X,x ) to (Y,y ) in
0 0 0
Top are continuous maps f: X →Y such that f(x )=y .
∗ 0 0
Definition (recollection). Two maps f, g: X →Y are called homotopic (f (cid:39)g)
if there exists a homotopy between them, i.e. a map H: X ×[0,1] → Y with
H(x,0) = f(x) and H(x,1) = g(x) for all x ∈ X. We call two spaces X and Y
homotopy equivalent if there are maps f: X → Y and g: Y → X such that
g◦f (cid:39)id and f ◦g (cid:39)id .
X Y
9
5.1. Cones, mapping cones, and suspensions.
Definition. Let X be a space. Its (unreduced) cone is the space
CX =X×[0,1]/∼,
where (x,1) ∼ (x(cid:48),1) for all x, x(cid:48) ∈ X. If x is a fixed base point of X, we also
0
denote its reduced cone by CredX; it is defined by
CX =X×[0,1]/sim,
where (x,1)∼(x(cid:48),1) as before but also (x ,t)=(x ,t(cid:48)) for all t, t(cid:48) ∈[0,1].
0 0
Lemma5.1. Amapf: X →Y ishomotopictoaconstantmap(“null-homotopic”)
iff it extends to a map f˜: CX →Y from the unreduced cone on X to Y.
A pointed map f: (X,x )→(Y,y ) is homotopic to the constant map with value
0 0
y via a homotopy that does not move x iff it extends to a map f˜: CredX → Y
0 0
from the reduced cone on X to Y.
Definition. Given a map f: A→X, define its (unreduced) mapping cone by
C =(A×[0,1](cid:116)X)/∼,
f
where (a,1)∼(a(cid:48),1) for all a, a(cid:48) ∈A and (a,0)∼f(a) for a∈A. Similarly, if f is
a pointed map with f(a )=x , the reduced mapping cone Cred is obtained by
0 0 f
adding
(a ,t)∼(a ,t(cid:48))∼x
0 0 0
to the equivalence relation, for all t, t(cid:48) ∈[0,1].
Lemma 5.2. Let f: A → X, g: X → Y be maps. Then g extends to g˜: C → Y
f
iff the composite g◦f is homotopic to a constant map.
If all maps are pointed then g extends to g˜: Cred →Y iff the composite g◦f is
f
homotopic to the constant map with value y via a homotopy that does not move
0
x .
0
Definition. The unreduced suspension SX of a space X is the unreduced
mapping cone of the unique map X → ∗; the reduced suspension ΣX of a
pointed space X is the reduced mapping cone of the unique pointed map X →∗.
Remark 5.3. For “good” spaces X and base points x ∈ X, the quotient maps
0
CX →CredX,SX →ΣX,and,forbasedmapsA→X,C →Cred,arehomotopy
f f
equivalences. “Good” here means “well-pointed”, which is implied for instance if
x has a contractible neighborhood in X.
0
5.2. The Eilenberg-Steenrod axioms. Let R be a ring, A an R-module, and
H : Top→Mod
n R
be a sequence of functors. We write H˜ (X) = ker(H (X)→H (∗)), where the
n n n
map is induced by the unique map X →∗.
Then (H ) is called a homology theory with coefficients in A if the
n n∈Z
following axioms hold:
homotopy: if f (cid:39)g then H (f)=H (g) for all n∈Z.
n n
additivity: if X = (cid:96) X then (cid:76) H (X ) ∼= H (X); the isomorphism
i∈I i i∈I n i n
is given by the canonical inclusions X (cid:44)→X.
i
(cid:40)
0; n(cid:54)=0
dimension: H (∗)= Inparticular,H (X)∼=H˜ (X)forn(cid:54)=0.
n n n
A; n=0.
10
exactness: Let f: A→X be a map and g: X →C be the standard inclu-
f
sion. Then there is a natural long exact sequence
···→H (A)−f→∗ H (X)−g→∗ H˜ (C )→H (A)→···
n n n f n−1
Mayer-Vietoris: Let X = U ∪V, where U and V are open subsets of X,
and Z =U ∩V. Then there is a long exact sequence
···→H (Z)−i−∗−−−j→∗ H (U)⊕H (V)−p−∗−+−q→∗ H (X)→H (Z)→··· ,
n n n n n−1
where the map i: Z (cid:44)→ U, j: Z (cid:44)→ V, p: U (cid:44)→ X, q: V (cid:44)→ X are all the
standard inclusions.
Theorem 5.4. For every ring R and every R-module A, there exists (up to equiv-
alence of functors) precisely one homology theory with coefficients in A.
5.3. Beginning calculations. For simplicity, let R=Z, A=Z.
(cid:40)
0; n(cid:54)=0
Lemma 5.5. If X is discrete then H (X)∼=
n (cid:76)
Z; n=0.
x∈X
Lemma 5.6. Denote by Sk the standard k-dimensional sphere. Then
(cid:40)
0; n(cid:54)=k
H˜ (Sk)∼=
n
Z; n=k.
Lemma 5.7. For any pointed space X, H (ΣX)∼=H˜ (X).
n+1 n
Lemma 5.8. Let Dn+1 be the (n+1)-dimensional disk, which has Sn as bound-
ary. There is no continuous function Dn+1 → Sn which is the identity, or even
homotopic to the identity, on Sn.
Corollary 5.9 (Brouwer’s fixedpointtheorem). Every continuous self-map of Dn
has a fixed point.
5.4. Mapping degrees. A map f: Sn →Sn gives a homomorphism of homology
groupsH (Sn)∼=Z,soit’smultiplicationbyanumberd,calledthemappingdegree
n
of f, deg(f).
Lemma5.10. Iff: Sn →Sn ishomotopictoaconstantmapthendeg(f)=0. (cid:3)
Lemma 5.11. deg(id)=1
Lemma 5.12. deg(f ◦g)=deg(f)deg(g).
Lemma 5.13. If f ∈O(n+1) then deg(f)=det(f).
Corollary 5.14. The map x(cid:55)→−x on Sn has degree (−1)n+1. (This map is called
the antipodal map.)
Corollary 5.15. If f: Sn →Sn has no fixed points then degf =(−1)n+1.
Theorem 5.16 (Hairy ball theorem). Let n be even and f: Sn → Rn+1 be a
continuous map such that f(x)⊥x for all x. Then f(x)=0 for some x∈Sn.