Chapter 29

Homological algebra

29.1 Homology of a chain complex

A chain complex C∗ is a sequence of abelian groups and homomor- phisms

/ / / / C∗ : ··· Cn+1 Cn Cn−1 ··· ∂n+1 ∂n

where (i) the elements of Cn are called n-chains, and / (ii) the homomorphisms ∂n : Cn Cn−1 are called boundary homo- morphisms.

These are required to satisfy the condition

∂n+1 ◦ ∂n =0 for each integer n. We shall only consider those with Cn =0for n<0. Consequently, ∂n =0for n ≤ 0. The n-cycles are those elements in Cn whose boundaries are zero: / Zn(C):=ker(∂n : Cn Cn−1).

In particular, Z0 = C0 On the other hand, the n-boundaries are those elements of Cn which are images of elements of Cn+1 under the boundary homomorphism ∂n+1: 1002 Homological algebra

/ Bn(C):=Im(∂n+1 : Cn+1 Cn).

From the condition ∂n+1 ◦ ∂n =0,wehaveIm ∂n+1 ⊆ ker ∂n for each integer n. The homology of the chain complex C measures its deviation from exactness: for each integer n,

Hn(C∗):=Zn(C)/Bn(C)=ker∂n/Im ∂n+1.

In particular, Hn(C)=0for n<0. Theorem 29.1. Every short exact sequence of chain complexes gives rise to a long exact sequence in homology: if 0 /C /C /C /0 f g

is an exact sequence of chain complexes, then the sequence

/  / /  /  / / ··· Hn+1(C ) Hn(C) Hn(C ) Hn(C ) Hn−1(C) ··· dn+1 f∗ g∗ dn

is exact.

Proposition 29.2. A commutative diagram

0 /C /C /C /0

   0 /E /E /E /0

of short exact sequences of chain complexes induces a commutative diagram of exact homology sequences

/  / /  /  / / ··· Hn+1(C ) Hn(C) Hn(C ) Hn(C ) Hn−1(C) ···

     /  / /  /  / / ··· Hn+1(E ) Hn(E) Hn(E ) Hn(E ) Hn−1(E) ··· 29.2 Cohomology of a chain complex 1003

29.2 Cohomology of a chain complex

A cochain complex C∗ is a sequence of abelian groups and homomor- phisms

n−1 n C∗ : ··· /Cn−1 δ /Cn δ /Cn+1 /···

where (i) the elements of Cn are called n-cochains, and (ii) the homomorphisms δn : Cn /Cn+1 are called coboundary homo- morphisms. These are required to satisfied the condition δn ◦ δn−1 =0

for each integer n. The elements of ker δn are called the n-cocycles, and those of Im δn−1 the n-coboundaries. The n-th cohomology is the quotient ker δn : Cn /Cn+1 Hn(C∗):= . Im δn−1 : Cn−1 /Cn

Example 29.1. Let C∗ be a chain complex. For each integer n, let Cn := Hom(Cn, Z) be the dual of Cn. These define a cochain complex C∗ with coboundary homomorphism δn : Cn /Cn+1 given by n = ∗ δ ∂n+1. 1004 Homological algebra Chapter 30

Tensor and torsion products

30.1 Tensor product

Let A and B be abelian groups. The tensor product A ⊗ B is defined by the following universal property: there is a bilinear map τ : A × B /A ⊗ B such that for every bilinear map φ : A × B /C, there is a unique homomorphism A ⊗ B /C such that

A × BBCφ /C ??  ??  ?  ??  τ ??  f ?  A ⊗ B is commutative. A ⊗ B is generated by elements of the form a ⊗ b, a ∈ A, b ∈ B, subject to the relations (i) a ⊗ 0=0⊗ b =0for a ∈ A, b ∈ B; (ii) n(a ⊗ b)=(na) ⊗ b = a ⊗ (nb) for a ∈ A, b ∈ B, and n ∈ Z; (iii) (a + a) ⊗ b = a ⊗ b + a ⊗ b and (iv) a ⊗ (b + b)=a ⊗ b + a ⊗ b.

30.1.1 Basic properties (1) A ⊗ B ≈ B ⊗ A. (2) Given f : A /B and g : A /B, there is an induced map f ⊗ g : A ⊗ B /A ⊗ B given by f ⊗ g(a ⊗ b)=f(a) ⊗ g(b). 1006 Tensor and torsion products

(3) (f  ◦ f) ⊗ (g ◦ g)=(f  ⊗ g) ◦ (f ⊗ g). (4) If A ≈ A1 ⊕ A2 ⊕···⊕An, then

A ⊗ B ≈ (A1 ⊗ B) ⊕ (A2 ⊗ B) ⊕···⊕(An ⊗ B).

(6) If A is a free abelian group with basis {ai} and B is a free abelian group with basis {bj}, then A ⊗ B is a free abelian group with basis {ai ⊗ bj}. (7) If i : AtoA and j : B /B are inclusions, then A ⊗ B (A/A) ⊗ (B/B) ≈ . Im (i ⊗ ι)+Im(ι ⊗ j)

Theorem 30.1. (1) Z ⊗ A ≈ A. (2) Zm ⊗ Zn ≈ Zgcd(m,n). Proposition 30.2. ⊗ is a right-exact functor: if

f g A /B /C /0 is exact, then for any abelian group D,

f⊗ι g⊗ι A ⊗ D /B ⊗ B /C ⊗ D /0 is also exact.

30.2 Torsion product

However, ⊗ is in general not left-exact. This means that if 0 A/ B/ C/ is exact, the tensored sequence 0 /A ⊗ D /B ⊗ D /C ⊗ D may not be exact. This leads to the notion of torsion product. Given an abelian group A and a free resolution

/ f1 / f0 / / ··· F1 F0 A 0,

for any abelian group B, the kernel of f1 ⊗ ι in the sequence f1⊗ι / / / F1 ⊗ B F0 ⊗ B A ⊗ B 0

is independent of the choose of the free abelian groups F0 and F1.Itis denoted by Tor(A, B) and is called the torsion product of A and B. 30.3 Homology with coefficients 1007

30.2.1 Basic properties of torsion product (1) If 0 /A /B /C /0 is exact, then for arbitrary abelian group D, 0 Tor/ (A, D) Tor/ (B,D) Tor/ (C, D) A/⊗D B/ ⊗D C/⊗D 0

is exact. (2) The torsion product is an additive functor. (3) If A is torsion-free, then Tor(A, B)=0for any B. (4) Tor(A, B) ≈ Tor(B,A). (5) Tor(Zn,B)={b ∈ B : nb =0}.

30.3 Homology with coefficients

Let C∗ be a chain complex, and G an abelian group. The homology of C∗ with coefficients G is the homology of the chain complex / / / / C∗ ⊗ G : ··· Cn+1 ⊗ G Cn ⊗ G Cn−1 ⊗ G ··· ∂n+1⊗ι ∂n⊗ι

Thus,

ker ∂n+1 ⊗ ι Hn(C∗; G):= . Im ∂n ⊗ ι

THEOREM (Universal coefficient theorem). Let C∗ be a free chain complex of abelian groups and G an abelian group, then

Hn(C; G) ≈ Hn(C) ⊗ G ⊕ Tor(Hn−1(C),G). In fact, this follows from an exact sequence

/ α / / / 0 Hn(C) ⊗ G Hn(C ⊗ G) Tor(Hn−1(C,G) 0, where α([c] ⊗ g)=[c ⊗ g] In particular, if G is torsion free, then α is an isomorphism between Hn(C∗; G)=Hn(C ⊗ G) and Hn(C) ⊗ G. 1008 Tensor and torsion products Chapter 31

The Hom and Ext functors

31.1 The Ext functor

Let G be an abelian group. The functor Hom(−,G) is a contravariant f g functor, and is left exact. This means that if 0 /A /B /C /0 is exact, then

g∗ f ∗ 0 /Hom(C, G) /Hom(B,G) /Hom(A, G) is also exact. However, it may not be right exact. If

/ f1 / f0 / / F1 F0 A 0 is a free resolution of A, then for arbitrary abelian group B, the cokernel of ∗ f1 / Hom(F0,B) Hom(F1,B)

is independent of F0 and F1. We denote this by Ext(A, B) and call it the extension of A by B.

31.1.1 Basic properties of the ext functor (1) If 0 /A /B /C /0 is exact, then for arbitrary abelian group G,

0Hom(/Hom(C, G)Hom() /Hom(B,G))Hom(/Hom(A, G) ggg ggggg ggggg sggggg Ext(C, G))Ext(/Ext(B,G))Ext(/Ext(A, G)0) / 1010 The Hom and Ext functors

is exact. (2) The ext functor is additive. (3) If A is free (abelian), then Ext(A, B)=0for any B. (4) If B is divisible in the sense that for every b ∈ B and every integer n>0 there exists b ∈ B such that b = nb, then Ext(A, B)=0for every abelian group A (5) Ext(Zn,G)=G/nG. In particular, Ext(Zn, Z)=Zn. (6) Ext(Zm, Zn)=Zgcd(m,n). In particular if p is a prime, then Ext(A, Zp)=0for every abelian group A.

31.2 Cohomology with coefficients Z2

Let C∗ be a free chain complex. The cohomology of the cochain complex C∗ with n C := Hom(Cn, Z2) satisfies n ∗ H (C )=Hom(Hn(C∗), Z2) for each integer n ≥ 0. Chapter 32

Cellular structures of projective spaces

32.1 Relative n-cells

Let f : Sn−1 /Y be a given map. The mapping cone of f is the space n n Z := Y ∪f e := Y ∪ D /(f(x)=x). We say that Z is obtained from Y by adjoining an n-cell via the map f. From the exact homotopy sequence of the pair (Z, Y ),wehave n πk(Y ∪f e ,Y) ≈ πk(Y ) for k

/ n Proposition 32.1. A map g : Y Z can be extended to Y ∪f e if and only if the composite g ◦ f is nullhomotopic.

32.2 Cellular structure of the real projective spaces

Let p : Sn /RPn be the double covering map of the real projec- (En Sn−1) ≈ tive space. This restricts to a relative homeomorphism +, (RPn, RPn−1). It follows that n n−1 n RP = RP ∪p e . Since RP1 = e0 ∪ e1, it follows by induction that RPn has a cellular structure consisting of one cell in each dimension through n: RPn = e0 ∪ e1 ∪···∪en. 1102 Cellular structures of projective spaces

32.3 Cellular structure of the complex projective spaces

Let q : S2n−1 /CPn−1 be the quotient map. The mapping cone of q is homeomorphic to the complex projective n-space. Thus,

n n−1 2n CP = CP ∪q e ,

and by induction, the complex projective n-space has a cell in each even dimension through 2n:

CPn = e0 ∪ e2 ∪···∪e2k ∪···∪en.

32.4 CW complexes

A CW complex is a space X with a filtration X0 ⊂ X1 ⊂···⊂Xk ⊂···= Xk k≥0

into skeleta of cells: each X k is a union of cells of dimensions ≤ k. k+1 k If X = X , it is obtained from Xk by adjoining a (finite) num- ber of (k +1)-cells. This means that there is a disjoint union Dk+1 = nk+1 Dk+1 : / k k+1 = k ∪ i=1 i and a map αk Dk+1 X such that X X αi Dk+1. The topology of X is the union topology of the skeleta. A subset Y ⊂ X is closed if and only if Y ∩ X n is closed in X n for every n ≥ 0.

32.4.1 Basic properties of CW complexes A CW complex X is (i) Hausdorff and normal, (ii) paracompact, (iii) locally path-connected. If X is connected, then it is path-connected. (2) Every compact subset of a CW complex X lies inside X n for some n. 32.4 CW complexes 1103

32.4.2 n-equivalences / A map f :(X, x0) (Y,y0) is an n-equivalence if / f∗ : πq(X, x0) πq(Y,y0) is an isomorphism for q ≤ n − 1 and is an epimorphism for q = n. It is a weak homotopy equivalence if it is an n-equivalence for every n ≥ 1. Theorem 32.2 (J. H. C. Whitehead). Every weak homotopy equiva- lence between CW complexes is a homotopy equivalence.

32.4.3 Cellular approximation theorem A CW complex A with filtration A0 ⊂ A1 ⊂···⊂Ak ⊂···

is a subcomplex of X if An ⊂ Xn for each n.ACW(X, A) pair con- sists of a CW complex X and a subcomplex A. A map f :(X, A) (/Y,B) between CW pairs is cellular if f[X n ∪ A] ⊂ Y n ∪ B for every n ≥ 0. THEOREM. Let g :(X, A) /(Y,B) be a map between CW pairs. There exists a cellular map f :(X, A) /(Y,B) such that f ∼ g rel A. Corollary 32.3. Let X be a CW complex with a single 0-cell and all other cells with dimensions >n. Then, X is n-connected.

n n / Proof. X = {∗}. Given a map f :(S ,x0) (X, x0), there is a n / n / n cellular approx g :(S ,x0) (X ,x0) (X, x0). Since X = {x0}, such a map is nullhomotopic. Therefore, f is nullhomotopic, and πq(X, x0)=0for q ≤ n. Corollary 32.4. Let X be a pointed CW complex. Its n-th suspension ΣnX is an (n − 1)-connected CW complex. 1104 Cellular structures of projective spaces Chapter 33

Cellular homology

33.1 The cellular chain complex of a CW complex

Let X be a CW complex. Define the cellular chain complex C∗(X) by 1 taking for Cn(X) the free abelian group generated by the n-cells of X. / The boundary homomorphism ∂n : Cn(X) Cn−1(X) is defined as n : Sn−1 / n−1 follows. Each n-cell eα has an attaching map fα X . If we collapse the (n − 2)-skeleton into a point, we obtain a bouquet of (n − 1)-spheres: n−1 n−2 = Sn−1 X /X γ . γ

Therefore, for each β, there is an integer dα,β which is the degree of the composite Sn−1 fα / n−1 / n−1 n−2 = Sn−1 /Sn−1 α X X /X γ β . γ By defining ( n)= n−1 ∂ eα dα,βeβ β

and extension by linearity, we have a chain complex C∗(X) since for each integer n ≥ 1, ∂n ◦ ∂n−1 =0.

The homology of C∗(X) is called the cellular homology of X, which we denote by H∗(X).

1For n<0, take Cn(X)=0so that the boundary homomorphism ∂0 =0. 1106 Cellular homology

33.2 Cellular homology of the sphere Sn

The sphere Sn has a cellular decomposition

Sn = e0 ∪ en.

n Therefore, Cq(S )=Z for q =0,n, and is zero otherwise. If n>1, there are no cells in consecutive dimensions and the boundary homo- morphism ∂n =0. It follows that Z,q=0,n; H (Sn)= q 0, otherwise.

The same is true for S1, by a separate treatment.

33.3 Cellular homology of CPn

The complex projective space CPn has cellular decomposition

CPn = e0 ∪ e2 ∪···∪e2n

consisting of one cell in each even dimension. Therefore, Z,qeven, C (CPn)= q 0,qodd.

The boundary homomorphisms are clearly trivial since there are no cells in consecutive dimensions. It follows that Z,qeven, H (CPn)= q 0,qodd.

33.4 Homology of real projective spaces

The real projective space RPn has one cell in each dimension q = 0, 1, ..., n: RPn = e0 ∪ e1 ∪···∪en. 33.4 Homology of real projective spaces 1107

q Therefore Cq is a free abelian group with generator e . The boundary homomorphism ∂q is the degree of the map Sq−1 /RPq−1 /RPq−1/RPq−2 = Sq−1,

which is the same as

ρq Sq−1 /RPq−1 /O(q) /Sq−1.

This is well known to have degree 1+(−1)q. Therefore, the boundary map is given by 2eq−1, if q is even and nonzero, ∂ (eq)= q 0, otherwise.

/ / / / / / / ··· C2k C2k−1 C2k−2 ··· C1 C0 0

··· 0 /ZZZ 2 /ZZZ 0 /Z 2 /··· 2 /ZZZ0 /Z 0 /0

From this, we conclude n (i) for even values of q ≤ n, Hq(RP )=0, n (ii) for odd values of q

33.4.1 Mod 2 homology of RPn The mod 2 homology of a CW complex X is the homology of the chain complex C∗(X) ⊗ Z2, with chain groups Cn(X) ⊗ Z2 and boundary homomorphisms are ∂n × ι. 1108 Cellular homology

For the real projective space RPn, (i) each chain group Cq = Z2 for 0 ≤ q ≤ n, (ii) each boundary homomorphism ∂q is trivial. It follows that Z , 0 ≤ q ≤ n, H (RPn; Z )= 2 q 2 0, otherwise. Chapter 34

Mod 2 cohomology of the RPn

34.1 Multiplicative structure

The mod 2 cellular cochain complex of RPn is given by

n ∗ n k C (RP ) ⊗ Z2 = [e ] · Z2 k=0

with zero differentials, where [ek] is the mod 2 dual of the chain (ek), k =0,...,n. The mod 2 cohomology of RPn is

n ∗ n k H (RP ; Z2)= [e ] · Z2. k=0

The mod 2 cohomology of a CW complex has a multiplicative struc- ture. For RPn, this is given by [eh+k],h+ k ≤ n, [eh] · [ek]= 0, otherwise.

Writing [e1]=w, we translate this as

∗ n n+1 1 n H (RP ; Z2)=Z2[w]/(w ),w∈ H (RP ; Z2). 1110 Mod 2 cohomology of the RPn

34.2 Applications

Lemma 34.1. Let f : Sm −→ Sn be a map satisfying f(−x)=−f(x), then it induces a map g : RPm /RPn such that

g∗(u)=v,

1 n 1 m where u ∈ H (RP ; Z2) and v ∈ H (RP ; Z2) are 1-dimensional cohomology generators.

Theorem 34.2 (Borsuk Ulam Theorem). If m>n, there is no contin- uous map f : Sm −→ Sn satisfying f(−x)=−f(x).

Proof. Consider the map g : RPm /RPn induced by f. By the above lemma g∗(u)=v for the cohomology generators u and v in the respec- tive projective spaces. However, if m>n,

0=g∗(un)=(g∗(u))n = vn =0,

a contradiction.

Theorem 34.3 (Hopf-Stiefel theorem). If there is a nonsingular bilin- ear map f : Rr × Rs −→ Rn, then n ≡ 0mod2for n − s

Proof. The bilinear map induces a map

g : RPr−1 × RPs−1 /RPn−1.

The mod 2 cohomology of the product RPr−1 × RPs−1 is the tensor product of the cohomology of the factor spaces:

∗ r−1 s−1 ∗ r−1 ∗ s−1 H (RP × RP ; Z2)=H (RP ; Z2) ⊗ H (RP ; Z2) r s =Z2[u]/(u ) ⊗ Z2[v]/(v ), where u and v are 1-dimensional cohomology generators. The induced map of g in cohomology satisfies

g∗(w)=u ⊗ 1+1⊗ v. 34.3 Steenrod squaring operations 1111

Since wn =0,wehave 0=g∗(wn) =(u ⊗ 1+1⊗ v)n n n = uk ⊗ vn−k k k=0 n = uk ⊗ vn−k. k n−s

34.3 Steenrod squaring operations

The mod 2 cohomology of a CW complex X also has a family of Steen- i k / k+i rod operations Sq : H (X; Z2) H (X; Z2), i =0, 1, 2,... sat- isfying 0 k / k (1) Sq : H (X; Z2) H (X; Z2) is the identity; k k 2 (2) if x ∈ H (X; Z2), Sq (x)=x ; (3) if k>dim x, Sqk(x)=0; k( ∪ )= i( ) ∪ j( ) (4) Cartan formula: Sq x y i+j=k Sq x Sq y . THEOREM. For the real projective space RPn, the Steenrod operations ∗ n on H (RP ; Z2) are given by k Sqh(xk)= xh+k,h≤ k. h