Cellular Homology

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.

To prove (b), consider the long exact sequence of the pair (Xⁿ, Xⁿ⁻¹) , which contains the segments

H_k+1(Xⁿ, Xⁿ⁻¹)

-→

^Hk(Xn−1)

-→

^Hk(Xn)

-→

^{Hk(Xn, Xn−1)}

If k is not equal to n or n− 1 then the outer two groups are zero by part (a), so we have isomorphisms H_k(Xⁿ⁻¹) ≈ H_k(Xⁿ) for k ≠ n, n − 1. Thus if k > n we have H_k(Xⁿ) ≈ Hk(Xⁿ⁻¹) ≈ Hk(Xⁿ⁻²)≈ ··· ≈ Hk(X⁰) = 0, proving (b). Further, if k < n then H_k(Xⁿ)≈ Hk(Xⁿ⁺¹)≈ ··· ≈ Hk(X^n+m) for all m≥ 0, proving (c) if X is finite-dimensional.

The proof of (c) when X is infinite-dimensional requires more work, and this can be done in two different ways. The more direct approach is to descend to the chain level and use the fact that a singular chain inX has compact image, hence meets only finitely many cells of X by Proposition A.1 in the Appendix. Thus each chain lies in a finite skeleton X^m. So a k cycle in X is a cycle in some X^m, and then by the finite-dimensional case of (c), the cycle is homologous to a cycle in Xⁿ if n > k , so i_∗:H_k(Xⁿ)

→

^Hk(X) is surjective. Similarly for injectivity, if a k cycle in Xⁿ bounds a chain in X , this chain lies in some X^m with m≥ n, so by the finite-dimensional case the cycle bounds a chain in Xⁿ ifn > k .

The other approach is more general. From the long exact sequence of the pair (X, Xⁿ) it suffices to show H_k(X, Xⁿ)= 0 for k ≤ n. Since H_k(X, Xⁿ)≈ eH_k(X/Xⁿ) , this reduces the problem to showing:

(∗) eH_k(X)= 0 for k ≤ n if the n skeleton of X is a point.

When X is finite-dimensional, (∗) is immediate from the finite-dimensional case of (c) which we have already shown. It will suffice therefore to reduce the infinite-dimensional case to the finite-infinite-dimensional case. This reduction will be achieved by stretchingX out to a complex that is at least locally finite-dimensional, using a special case of the ‘mapping telescope’ construction described in greater generality in§3.F.

R Consider X×[0, ∞) with its product cell structure,

where we give [0,∞) the cell structure with the integer points as 0 cells. Let T =S

iXⁱ×[i, ∞), a subcomplex

of X×[0, ∞). The figure shows a schematic picture of T with [0, ∞) in the hor-izontal direction and the subcomplexes Xⁱ×[i, i + 1] as rectangles whose size in-creases withi since Xⁱ⊂ Xⁱ⁺¹. The line labeledR can be ignored for now. We claim that T ' X , hence H_k(X) ≈ H_k(T ) for all k . Since X is a deformation retract of X×[0, ∞), it suffices to show that X×[0, ∞) also deformation retracts onto T . Let Y_i= T ∪ X×[i, ∞)

. ThenY_i deformation retracts ontoY_i+1sinceX×[i, i+1] defor-mation retracts ontoXⁱ×[i, i + 1] ∪ X×{i + 1} by Proposition 0.16. If we perform the deformation retraction of Y_i onto Y_i+1 during the t interval [1− 1/2ⁱ, 1− 1/2ⁱ⁺¹] , then this gives a deformation retraction f_t of X×[0, ∞) onto T , with points in Xⁱ×[0, ∞) stationary under ft for t≥ 1 − 1/2ⁱ⁺¹. Continuity follows from the fact

that CW complexes have the weak topology with respect to their skeleta, so a map is continuous if its restriction to each skeleton is continuous.

Recalling thatX⁰ is a point, letR⊂ T be the ray X⁰×[0, ∞) and let Z ⊂ T be the union of this ray with all the subcomplexes Xⁱ×{i}. Then Z/R is homeomorphic to W

iXⁱ, a wedge sum of finite-dimensional complexes with n skeleton a point, so the finite-dimensional case of (∗) together with Corollary 2.25 describing the homology of wedge sums implies that eH_k(Z/R)= 0 for k ≤ n. The same is therefore true for Z , from the long exact sequence of the pair(Z, R) , since R is contractible. Similarly, T /Z is a wedge sum of finite-dimensional complexes with (n+ 1) skeleton a point, since if we first collapse each subcomplex Xⁱ×{i} of T to a point, we obtain the infinite sequence of suspensionsSXⁱ ‘skewered’ along the rayR , and then if we collapse R to a point we obtain W

iΣXⁱ where ΣXⁱ is the reduced suspension of Xⁱ, obtained from SXⁱ by collapsing the line segmentX⁰×[i, i+1] to a point, so ΣXⁱ has(n+1) skeleton a point. Thus eH_k(T /Z)= 0 for k ≤ n + 1, and then the long exact sequence of the pair(T , Z) implies that eH_k(T )= 0 for k ≤ n, and we have proved (∗). tu

Let X be a CW complex. Using Lemma 2.34, portions of the long exact sequences for the pairs (Xⁿ⁺¹, Xⁿ) , (Xⁿ, Xⁿ⁻¹) , and (Xⁿ⁻¹, Xⁿ⁻²) fit into a diagram

-X X

-H_{n 1+}( ^{n 1+}, ⁿ)

n 1+

n 1+

n 1+

−−−−−→

^{H (n X X, )}

n n

n 1 n n 1- n 2

-n

H (_n Xⁿ) X

H (_n ) H (_n X)

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

X X H_{n 1}( , )

-n 1 - n 1

-X H_{n 1}( )

−−−−−→

. . . −−−−−→ ^{. . .}

≈ 0 0

0

∂

∂ j

j

d d

where d_n+1 and d_n are defined as the compositions j_n∂_n+1 and j_n−1∂_n, which are just ‘relativizations’ of the boundary maps ∂_n+1 and ∂_n. The composition d_nd_n+1 includes two successive maps in one of the exact sequences, hence is zero. Thus the horizontal row in the diagram is a chain complex, called thecellular chain complex of X since H_n(Xⁿ, Xⁿ⁻¹) is free with basis in one-to-one correspondence with the n cells of X , so one can think of elements of H_n(Xⁿ, Xⁿ⁻¹) as linear combinations of n cells of X . The homology groups of this cellular chain complex are called the cellular homology groups of X . Temporarily we denote them H^CW_n (X) .

T

heorem 2.35. H_n^CW(X)≈ H_n(X) .

P

roof: From the diagram above, H_n(X) can be identified with H_n(Xⁿ)/ Im ∂_n+1. Since j_n is injective, it maps Im∂_n+1 isomorphically onto Im(j_n∂_n+1) = Im d_n+1

and H_n(Xⁿ) isomorphically onto Im j_n = Ker ∂n. Since j_n−1 is injective, Ker∂_n = Kerd_n. Thus j_n induces an isomorphism of the quotient H_n(Xⁿ)/ Im ∂_n+1 onto

Kerd_n/ Im d_n+1. tu

Here are a few immediate applications:

(i) H_n(X)= 0 if X is a CW complex with no n cells.

(ii) More generally, if X is a CW complex with k n cells, then H_n(X) is generated by at mostk elements. For since H_n(Xⁿ, Xⁿ⁻¹) is free abelian on k generators, the subgroup Kerd_n must be generated by at most k elements, hence also the quotient Kerd_n/ Im d_n+1.

(iii) If X is a CW complex having no two of its cells in adjacent dimensions, then H_n(X) is free abelian with basis in one-to-one correspondence with the n cells of X . This is because the cellular boundary maps d_n are automatically zero in this case.

This last observation applies for example to CPⁿ, which has a CW structure with one cell of each even dimension 2k≤ 2n as we saw in Example 0.6. Thus

H_i(CPⁿ)≈

Z for i = 0, 2, 4, ··· , 2n 0 otherwise

Another simple example is Sⁿ×Sⁿ with n > 1 , using the product CW structure con-sisting of a 0 cell, twon cells, and a 2n cell.

It is possible to prove the statements (i)–(iii) for finite-dimensional CW complexes by induction on the dimension, without using cellular homology but only the basic results from the previous section. However, the viewpoint of cellular homology makes (i)–(iii) quite transparent.

Next we describe how the cellular boundary maps d_n can be computed. When n= 1 this is easy since the boundary map d₁:H₁(X¹, X⁰)

→

^H0(X0) is the same as the simplicial boundary map ∆₁(X)

→

^∆0(X) . In case X is connected and has only one 0 cell, thend₁ must be 0 , otherwiseH₀(X) would not beZ. When n > 1 we will show that d_n can be computed in terms of degrees:

Cellular Boundary Formula. d_n(eⁿ_α)=P

βd_αβeⁿ⁻¹_β where d_αβ is the degree of the map S_αⁿ⁻¹

→

^Xn−1

→

^Sβⁿ⁻¹ that is the composition of the attaching map of eⁿ_α with the quotient map collapsing Xⁿ⁻¹− eβⁿ⁻¹ to a point.

Here we are identifying the cells eⁿ_α and e_βⁿ⁻¹ with generators of the corresponding summands of the cellular chain groups. The summation in the formula contains only finitely many terms since the attaching map of eⁿ_α has compact image, so this image meets only finitely many cells eⁿ⁻¹_β .

To derive the cellular boundary formula, consider the commutative diagram

-n

n

-n 1 n 2

-X -X

-H (_n ⁿ, ^{n 1})

−−−−−→

−−−− −→

X X

H ( )

,

n 1

-n 1 - n 1

-H_{n 1}(X )

−−−−−−→

−−−−−−−−→

−−−→

≈

≈

≈

∂

j

q

d

∼ ∼

- n 1- n 2

-X X

H_{n 1}( ) H_{n 1-} (X^{n 1-} X^{n 2-} ,X^{n 2-} X^{n 2-} )

−−−−−→ −−−−−→

- n 1

-D D

H (_n ⁿ,^{∂ n})

−−−−−→

^{∂ H∼n 1-}

−−−−−→

^{( ) H∼n 1}

−−−−− →

^{(S )}

α α ∂Dαⁿ

α

α

β β

β∗

∗

q_∗

∗

∆

−−−−−→

Φ_α∗ ϕ

6 6 6

where:

Φα is the characteristic map of the cell e_αⁿ and ϕ_α is its attaching map.

q : Xⁿ⁻¹

→

^{Xn−1/Xn−2} is the quotient map.

q_β:Xⁿ⁻¹/Xⁿ⁻²

→

^Sβn−1 collapses the complement of the celleⁿ⁻¹_β to a point, the resulting quotient sphere being identified with S_βⁿ⁻¹= Dⁿ⁻¹_β /∂Dⁿ⁻¹_β via the char-acteristic mapΦβ.

∆αβ:∂D_αⁿ

→

^Sβn−1 is the composition q_βqϕ_α, in other words, the attaching map of e_αⁿ followed by the quotient map Xⁿ⁻¹

→

^Sβn−1 collapsing the complement of e_βⁿ⁻¹ inXⁿ⁻¹ to a point.

The map Φ_α∗ takes a chosen generator [Dⁿ_α]∈ Hn(Dⁿ_α, ∂Dⁿ_α) to a generator of the Z summand of H_n(Xⁿ, Xⁿ⁻¹) corresponding to eⁿ_α. Letting eⁿ_α denote this generator, commutativity of the left half of the diagram then gives d_n(eⁿ_α)= j_n−1ϕ_α∗∂[D_αⁿ] . In terms of the basis forH_n−1(Xⁿ⁻¹, Xⁿ⁻²) corresponding to the cells eⁿ⁻¹_β , the mapq_β∗

is the projection of H_n−1(Xⁿ⁻¹/Xⁿ⁻²) onto its Z summand corresponding to eⁿ⁻¹_β . Commutativity of the diagram then yields the formula ford_n given above.

E

xample 2.36. Let M_g be the closed orientable surface of genusg with its usual CW structure consisting of one 0 cell, 2g 1 cells, and one 2 cell attached by the product of commutators[a₁, b₁]··· [a_g, b_g] . The associated cellular chain complex is

0

---→

^Z

---→

^{d2 Z2g}

---→

^{d1 Z}

---→

⁰

As observed above,d₁ must be 0 since there is only one 0 cell. Also,d₂ is 0 because each a_i or b_i appears with its inverse in [a₁, b₁]··· [a_g, b_g] , so the maps ∆_αβ are homotopic to constant maps. Since d₁ and d₂ are both zero, the homology groups of M_g are the same as the cellular chain groups, namely, Z in dimensions 0 and 2, and Z^2g in dimension 1 .

E

xample 2.37. The closed nonorientable surface N_g of genus g has a cell structure with one 0 cell, g 1 cells, and one 2 cell attached by the word a²₁a²₂··· a²_g. Again d₁= 0, and d2:Z

→

^Zg is specified by the equationd₂(1)= (2, ··· , 2) since each ai

appears in the attaching word of the 2 cell with total exponent 2 , which means that each∆_αβis homotopic to the mapz

,

^z2, of degree 2 . Since d₂(1)= (2, ··· , 2), we have d₂ injective and hence H₂(N_g)= 0. If we change the basis for Z^g by replacing the last standard basis element (0,··· , 0, 1) by (1, ··· , 1), we see that H1(N_g) ≈ Z^g−1⊕Z2.

These two examples illustrate the general fact that the orientability of a closed connected manifold M of dimension n is detected by H_n(M) , which is Z if M is orientable and 0 otherwise. This is shown in Theorem 3.26.

E

xample 2.38: An Acyclic Space. Let X be obtained from S¹∨ S¹ by attaching two 2 cells by the words a⁵b⁻³ and b³(ab)⁻². Then d₂:Z²

→

^{Z2 has matrix }−3 1⁵⁻²

 , with the two columns coming from abelianizing a⁵b⁻³ and b³(ab)⁻² to 5a− 3b and −2a + b , in additive notation. The matrix has determinant −1, so d2 is an isomorphism and eH_i(X)= 0 for all i. Such a space X is called acyclic.

We can see that this acyclic space is not contractible by consideringπ₁(X) , which has the presentation

a, b |||| a⁵b⁻³, b³(ab)⁻²

. There is a nontrivial homomorphism from this group to the group G of rotational symmetries of a regular dodecahedron, sending a to the rotation ρ_a through angle 2π /5 about the axis through the center of a pentagonal face, and b to the rotation ρ_b through angle 2π /3 about the axis through a vertex of this face. The composition ρ_aρ_b is a rotation through angle π about the axis through the midpoint of an edge abutting this vertex. Thus the relations a⁵= b³= (ab)² defining π₁(X) become ρ_a⁵= ρ_b³= (ρ_aρ_b)²= 1 in G, which means there is a well-defined homomorphism ρ : π₁(X)

→

G sending a to ρ_a and b to ρ_b. It is not hard to see that G is generated by ρ_a and ρ_b, so ρ is surjective. With more work one can compute that the kernel of ρ is Z2, generated by the element a⁵= b³= (ab)², and this Z2 is in fact the center ofπ₁(X) . In particular, π₁(X) has order 120 since G has order 60.

After these 2 dimensional examples, let us now move up to three dimensions, where we have the additional task of computing the cellular boundary map d₃.

c c

c c

b b b b

a a a

a

c c

c c

b b b b

a a a

a

E

xample 2.39. A 3 dimensional torus T³ = S¹×S¹×S¹ can be constructed from a cube by identifying each pair of opposite square faces as in the first of the two figures. The second figure shows a slightly different pattern of

identifications of opposite faces, with the front and back faces now identified via a rotation of the cube around a horizontal left-right axis. The space produced by these identifications is the product K×S¹ of a Klein bottle and a circle. For both T³ and K×S¹ we have a CW structure with one 3 cell, three 2 cells, three 1 cells, and one 0 cell. The cellular chain complexes thus have the form

0

-→

^Z

---→

^{d3 Z3}

---→

^{d2 Z3}

---→

^{0 Z}

-→

⁰

In the case of the 3 torus T³ the cellular boundary map d₂ is zero by the same calculation as for the 2 dimensional torus. We claim that d₃ is zero as well. This amounts to saying that the three maps∆αβ:S²

→

^S2corresponding to the three 2 cells

have degree zero. Each ∆αβ maps the interiors of two opposite faces of the cube homeomorphically onto the complement of a point in the target S² and sends the remaining four faces to this point. Computing local degrees at the center points of the two opposite faces, we see that the local degree is+1 at one of these points and

−1 at the other, since the restrictions of ∆αβ to these two faces differ by a reflection of the boundary of the cube across the plane midway between them, and a reflection has degree−1. Since the cellular boundary maps are all zero, we deduce that H_i(T³) is Z for i = 0, 3, Z³ fori= 1, 2, and 0 for i > 3.

For K×S¹, when we compute local degrees for the front and back faces we find that the degrees now have the same rather than opposite signs since the map∆_αβ on these two faces differs not by a reflection but by a rotation of the boundary of the cube.

The local degrees for the other faces are the same as before. Using the lettersA , B , C to denote the 2 cells given by the faces orthogonal to the edgesa , b , c , respectively, we have the boundary formulas d₃e³ = 2C , d₂A = 2b , d₂B = 0, and d₂C = 0. It follows that H₃(K×S¹)= 0, H₂(K×S¹)= Z⊕Z₂, andH₁(K×S¹)= Z⊕Z⊕Z₂.

Many more examples of a similar nature, quotients of a cube or other polyhedron with faces identified in some pattern, could be worked out in similar fashion. But let us instead turn to some higher-dimensional examples.

E

xample 2.40: Moore Spaces. Given an abelian group G and an integer n≥ 1, we will construct a CW complexX such that H_n(X)≈ G and eH_i(X)= 0 for i ≠ n. Such a space is called aMoore space, commonly written M(G, n) to indicate the dependence on G and n . It is probably best for the definition of a Moore space to include the condition that M(G, n) be simply-connected if n > 1 . The spaces we construct will have this property.

As an easy special case, when G= Z_m we can take X to be Sⁿ with a cell eⁿ⁺¹ attached by a mapSⁿ

→

^{Snof degree}m . More generally, any finitely generated G can be realized by taking wedge sums of examples of this type for finite cyclic summands ofG , together with copies of Sⁿ for infinite cyclic summands of G .

In the general nonfinitely generated case letF

→

G be a homomorphism of a free abelian groupF onto G , sending a basis for F onto some set of generators of G . The kernelK of this homomorphism is a subgroup of a free abelian group, hence is itself free abelian. Choose bases {x_α} for F and {y_β} for K , and write y_β=P

αd_βαx_α. Let Xⁿ=W

αS_αⁿ, so H_n(Xⁿ)≈ F via Corollary 2.25. We will construct X from Xⁿ by attaching cells eⁿ⁺¹_β via maps f_β:Sⁿ

→

^Xn such that the composition of f_β with the projection onto the summand S_αⁿ has degree d_βα. Then the cellular boundary map d_n+1 will be the inclusionK

>

F , hence X will have the desired homology groups.

The construction of f_β generalizes the construction in Example 2.31 of a map Sⁿ

→

^Sn of given degree. Namely, we can let f_β map the complement of P

α|dβα|

disjoint balls in Sⁿ to the 0 cell of Xⁿ while sending |dβα| of the balls onto the summandS_αⁿ by maps of degree+1 if d_βα> 0 , or degree −1 if d_βα< 0 .

E

xample 2.41. By taking a wedge sum of the Moore spaces constructed in the preced-ing example for varypreced-ing n we obtain a connected CW complex with any prescribed sequence of homology groups in dimensions 1, 2, 3,···.

E

xample 2.42: Real Projective Space RPⁿ. As we saw in Example 0.4,RPⁿ has a CW structure with one celle^kin each dimensionk≤ n, and the attaching map for e^kis the 2 sheeted covering projection ϕ : S^k−1

→

^RPk−1. To compute the boundary map d_k we compute the degree of the composition S^{k−1 ϕ}

---→

^{RPk−1 q}

---→

^{RPk−1/RPk−2= Sk−1,}

with q the quotient map. The map qϕ is a homeomorphism when restricted to each component of S^k−1− S^k−2, and these two homeomorphisms are obtained from each other by precomposing with the antipodal map of S^k−1, which has degree (−1)^k. Hence degqϕ= deg¹1+ deg(−¹1)= 1 + (−1)^k, and sod_k is either 0 or multiplication by 2 according to whetherk is odd or even. Thus the cellular chain complex forRPⁿ is

0

-→

^Z

---→

^{2 Z}

---→

^{0 ···}

---→

^{2 Z}

---→

^{0 Z}

---→

^{2 Z}

---→

^{0 Z}

-→

^{0 ifn is even}

0

-→

^Z

---→

^{0 Z}

---→

^{2 ···}

---→

^{2 Z}

---→

^{0 Z}

---→

^{2 Z}

---→

^{0 Z}

-→

^{0 ifn is odd}

From this it follows that

H_k(RPⁿ)=





Z fork= 0 and for k = n odd Z₂ fork odd, 0 < k < n 0 otherwise

E

xample 2.43: Lens Spaces. This example is somewhat more complicated. Given an integerm > 1 and integers `<sub>1</sub>,··· , `nrelatively prime tom , define the lens space L= L_m(`<sub>1</sub>,··· , `_n) to be the orbit space S²ⁿ⁻¹/Z_mof the unit sphereS²ⁿ⁻¹⊂ Cⁿ with the action ofZ_m generated by the rotation ρ(z₁,··· , z_n)= (e^{2π i`1/m</sup>z<sub>1</sub>,··· , e<sup>2π i`n/m}z_n) , rotating the j^th C factor of Cⁿ by the angle 2π `_j/m . In particular, when m= 2, ρ is the antipodal map, so L= RP²ⁿ⁻¹ in this case. In the general case, the projection S²ⁿ⁻¹

→

L is a covering space since the action ofZmonS²ⁿ⁻¹ is free: Only the identity element fixes any point of S²ⁿ⁻¹ since each point of S²ⁿ⁻¹ has some coordinate z_j nonzero and thene^{2π ik`j/m}z_j≠ z_j for 0< k < m , as a result of the assumption that

`_j is relatively prime tom .

We shall construct a CW structure on L with one cell e^k for eachk≤ 2n − 1 and show that the resulting cellular chain complex is

0

-→

^Z

---→

^{0 Z}

---→

^{m Z}

---→

^{0 ···}

---→

^{0 Z}

---→

^{m Z}

---→

^{0 Z}

-→

⁰

with boundary maps alternately 0 and multiplication bym . Hence

H_k L_m(`<sub>1</sub>,··· , `n)

=





Z fork= 0, 2n − 1

Zm fork odd, 0 < k < 2n− 1 0 otherwise

To obtain the CW structure, first subdivide the unit circle C in the n^th C factor of Cⁿ by taking the points e^{2π ij/m} ∈ C as vertices, j = 1, ··· , m. Joining the j^th vertex of C to the unit sphere S²ⁿ⁻³⊂ Cⁿ⁻¹ by arcs of great circles in S²ⁿ⁻¹ yields a (2n− 2) dimensional ball B²ⁿ⁻²j bounded by S²ⁿ⁻³. Specifically, B_j²ⁿ⁻² consists of the points cosθ (0,··· , 0, e^{2π ij/m})+sin θ (z1,··· , z_n−1, 0) for 0≤ θ ≤ π/2. Similarly, joining the j^th edge ofC to S²ⁿ⁻³ gives a ball B²ⁿ⁻¹_j bounded byB_j²ⁿ⁻² and B_j+1²ⁿ⁻², subscripts being taken mod m . The rotation ρ carries S²ⁿ⁻³ to itself and rotates C by the angle 2π `<sub>n</sub>/m , hence ρ permutes the B<sup>2n−2</sup><sub>j</sub> ’s and the B<sub>j</sub><sup>2n−1</sup>’s. A suitable power of ρ , namely ρ<sup>r</sup> where r `_n ≡ 1 mod m, takes each Bj²ⁿ⁻² and B_j²ⁿ⁻¹ to the next one. Since ρ^r has order m , it is also a generator of the rotation group Zm, and hence we may obtainL as the quotient of one B_j²ⁿ⁻¹ by identifying its two facesB_j²ⁿ⁻² and B_j+1²ⁿ⁻² together via ρ^r.

In particular, when n= 2, Bj²ⁿ⁻¹ is a lens-shaped 3 ball andL is obtained from this ball by identifying its two curved disk faces viaρ^r, which may be described as the composition of the reflection across the plane con-taining the rim of the lens, taking one face of the lens to the other, followed by a rotation of this face through the angle 2π `/m where `= r `₁. The figure illustrates the

case(m, `)= (7, 2), with the two dots indicating a typical pair of identified points in the upper and lower faces of the lens. Since the lens spaceL is determined by the rota-tion angle 2π `/m , it is conveniently written L<sub>`/m</sub>. Clearly only the modm value of ` matters. It is a classical theorem of Reidemeister from the 1930s thatL_{`/m</sub> is homeo-morphic toL<sub>`}0/m⁰ iffm⁰= m and `<sup>0</sup>≡ ±`^±1 modm . For example, when m= 7 there are only two distinct lens spacesL_1/7 and L_2/7. The ‘if’ part of this theorem is easy:

Reflecting the lens through a mirror shows that L_{`/m</sub>≈ L<sub>−`/m}, and by interchanging the roles of the two C factors of C² one obtainsL<sub>`/m</sub> ≈ L`⁻¹/m. In the converse di-rection, L_{`/m</sub>≈ L<sub>`}0/m⁰ clearly implies m= m⁰ since π₁(L_`/m)≈ Z_m. The rest of the theorem takes considerably more work, involving either special 3 dimensional tech-niques or more algebraic methods that generalize to classify the higher-dimensional lens spaces as well. The latter approach is explained in [Cohen 1973].

Returning to the construction of a CW structure on L_m(`<sub>1</sub>,··· , `n) , observe that the (2n− 3) dimensional lens space Lm(`<sub>1</sub>,··· , `_n−1) sits in L_m(`<sub>1</sub>,··· , `n) as the quotient of S²ⁿ⁻³, and L_m(`<sub>1</sub>,··· , `n) is obtained from this subspace by attaching two cells, of dimensions 2n− 2 and 2n − 1, coming from the interiors of B_j²ⁿ⁻¹ and its two identified faces B_j²ⁿ⁻² and B_j+1²ⁿ⁻². Inductively this gives a CW structure on L_m(`<sub>1</sub>,··· , `n) with one cell e^k in each dimension k≤ 2n − 1.

The boundary maps in the associated cellular chain complex are computed as follows. The first one, d_2n−1, is zero since the identification of the two faces of B²ⁿ⁻¹_j is via a reflection (degree −1) across Bj²ⁿ⁻¹ fixing S²ⁿ⁻³, followed by a

rota-tion (degree +1), so d_2n−1(e²ⁿ⁻¹) = e²ⁿ⁻²− e²ⁿ⁻² = 0. The next boundary map d_2n−2 takes e²ⁿ⁻² to me²ⁿ⁻³ since the attaching map for e²ⁿ⁻² is the quotient map S²ⁿ⁻³

→

^Lm(`<sub>1</sub>,··· , `_n−1) and the balls B_j²ⁿ⁻³inS²ⁿ⁻³which project down ontoe²ⁿ⁻³ are permuted cyclically by the rotation ρ of degree +1. Inductively, the subsequent boundary mapsd_k then alternate between 0 and multiplication by m .

Also of interest are the infinite-dimensional lens spacesL_m(`<sub>1</sub>, `₂,···) = S^∞/Zm

defined in the same way as in the finite-dimensional case, starting from a sequence of integers`<sub>1</sub>, `₂,··· relatively prime to m. The space L_m(`<sub>1</sub>, `₂,···) is the union of the increasing sequence of finite-dimensional lens spacesL_m(`<sub>1</sub>,··· , `n) for n= 1, 2, ···, each of which is a subcomplex of the next in the cell structure we have just con-structed, so L_m(`<sub>1</sub>, `₂,···) is also a CW complex. Its cellular chain complex consists of a Z in each dimension with boundary maps alternately 0 and m, so its reduced homology consists of a Z_m in each odd dimension.

In the terminology of§1.B, the infinite-dimensional lens space L_m(`<sub>1</sub>, `₂,···) is an Eilenberg–MacLane space K(Zm, 1) since its universal cover S^∞ is contractible, as we showed there. By Theorem 1B.8 the homotopy type of L_m(`<sub>1</sub>, `₂,···) depends only on m , and not on the `<sub>i</sub>’s. This is not true in the finite-dimensional case, when two lens spaces L<sub>m</sub>(`₁,··· , `<sub>n</sub>) and L<sub>m</sub>(`⁰₁,··· , `<sup>0</sup><sub>n</sub>) have the same homotopy type iff `₁··· `n≡ ±k<sup>n</sup>`⁰₁··· `⁰nmodm for some integer k . A proof of this is outlined in Exercise 2 in§3.E and Exercise 29 in §4.2. For example, the 3 dimensional lens spaces L_1/5 and L_2/5 are not homotopy equivalent, though they have the same fundamental group and the same homology groups. On the other hand,L_1/7 andL_2/7are homotopy equivalent but not homeomorphic.

Euler Characteristic

For a finite CW complex X , the Euler characteristic χ (X) is defined to be the alternating sum P

n(−1)ⁿc_n where c_n is the number of n cells of X , generalizing the familiar formula vertices − edges + faces for 2 dimensional complexes. The following result shows that χ(X) can be defined purely in terms of homology, and hence depends only on the homotopy type of X . In particular, χ (X) is independent of the choice of CW structure on X .

T

heorem 2.44. χ (X)=P

n(−1)ⁿrankH_n(X) .

Here therank of a finitely generated abelian group is the number ofZ summands when the group is expressed as a direct sum of cyclic groups. We shall need the following fact, whose proof we leave as an exercise: If 0

→

^A

→

^B

→

^C

→

0 is a short exact sequence of finitely generated abelian groups, then rankB= rank A + rank C .

P

roof of 2.44: This is purely algebraic. Let

0

-→

^Ck

---→

^{dk C}k−1

-→

^···

-→

^C1

---→

^{d1 C0}

-→

⁰

be a chain complex of finitely generated abelian groups, with cycles Z_n = Ker dn, boundaries B_n = Im d_n+1, and homology H_n = Z_n/B_n. Thus we have short exact sequences 0

→

^Zn

→

^Cn

→

^Bn−1

→

^{0 and 0}

→

^Bn

→

^Zn

→

^Hn

→

^{0 , hence}

rankC_n= rank Zn+ rank B_n−1 rankZ_n= rank Bn+ rank Hn

Now substitute the second equation into the first, multiply the resulting equation by (−1)ⁿ, and sum over n to get P

n(−1)ⁿrankC_n=P

n(−1)ⁿrankH_n. Applying this with C_n= H_n(Xⁿ, Xⁿ⁻¹) then gives the theorem. tu

For example, the surfacesM_g andN_g have Euler characteristics χ(M_g)= 2 − 2g and χ(N_g)= 2 − g . Thus all the orientable surfaces M_g are distinguished from each other by their Euler characteristics, as are the nonorientable surfaces N_g, and there are only the relations χ(M_g)= χ (N2g) .

Split Exact Sequences

Suppose one has a retractionr : X

→

^{A , so r i=11 wherei : A}

→

X is the inclusion.

The induced map i_∗:H_n(A)

→

^Hn(X) is then injective since r_∗i_∗=¹1 . From this it follows that the boundary maps in the long exact sequence for(X, A) are zero, so the long exact sequence breaks up into short exact sequences

0

-→

^Hn(A)

---→

^{i∗ H}n(X)

---→

^{j∗ H}n(X, A)

-→

⁰

The relation r_∗i_∗ = ¹1 actually gives more information than this, by the following piece of elementary algebra:

S

plitting

L

emma. For a short exact sequence 0

-→

^A

---→

^{i B}

---→

^{j C}

-→

^{0 of abelian}

groups the following statements are equivalent :

(a) There is a homomorphism p : B

→

A such that pi=¹1 :A

→

^{A .}

(b) There is a homomorphism s : C

→

B such that js=¹1 :C

→

^{C .}

(c) There is an isomorphism B ≈ A⊕^{C making}

a commutative diagram as at the right, where 0

−−−−−→

−−→

A

−−−−→

^B

−−−−→

−−−→

A C

−−−→

i

C

−−→

0

j

⊕

≈ the maps in the lower row are the obvious ones,

a

,

(a, 0) and (a, c)

,

^{c .}

If these conditions are satisfied, the exact sequence is said tosplit. Note that (c) is symmetric: There is no essential difference between the roles of A and C .

Sketch of Proof: For the implication (a)⇒ (c) one checks that the map B

→

^{A⊕C ,}

b

,

^{p(b), j(b)}
, is an isomorphism with the desired properties. For (b)⇒ (c) one uses instead the map A⊕^C

→

^{B , (a, c)}

,

^i(a)+ s(c). The opposite implications (c)⇒ (a) and (c) ⇒ (b) are fairly obvious. If one wants to show (b) ⇒ (a) directly, one can define p(b)= i⁻¹ b− sj(b)

. Further details are left to the reader. tu

Except for the implications(b)⇒ (a) and (b) ⇒ (c), the proof works equally well for nonabelian groups. In the nonabelian case, (b) is definitely weaker than (a) and (c), and short exact sequences satisfying (b) only determine B as a semidirect product of A and C . The difficulty is that s(C) might not be a normal subgroup of B . In the nonabelian case one defines ‘splitting’ to mean that (b) is satisfied.

In both the abelian and nonabelian contexts, ifC is free then every exact sequence 0

→

^A

---→

^{i B}

---→

^{j C}

→

0 splits, since one can defines : C

→

B by choosing a basis{c_α} for C and letting s(c_α) be any element b_α∈ B such that j(bα)= cα. The converse is also true: If every short exact sequence ending in C splits, then C is free. This is because for every C there is a short exact sequence 0

→

^A

→

^B

→

^C

→

^{0 with B free}

— choose generators forC and let B have a basis in one-to-one correspondence with these generators, then let B

→

C send each basis element to the corresponding gen-erator — so if this sequence 0

→

^A

→

^B

→

^C

→

^{0 splits,} C is isomorphic to a subgroup of a free group, hence is free.

From the Splitting Lemma and the remarks preceding it we deduce that a retrac-tionr : X

→

A gives a splitting H_n(X)≈ Hn(A)⊕H_n(X, A) . This can be used to show the nonexistence of such a retraction in some cases, for example in the situation of the Brouwer fixed point theorem, where a retraction Dⁿ

→

^Sn−1 would give an im-possible splitting H_n−1(Dⁿ) ≈ H_n−1(Sⁿ⁻¹)⊕^H_n−1^{(Dn, Sn−1}) . For a somewhat more subtle example, consider the mapping cylinder M_f of a degree m map f : Sⁿ

→

^Sn

with m > 1 . If M_f retracted onto the Sⁿ ⊂ Mf corresponding to the domain of f , we would have a split short exact sequence

−−−−−−→

^Z

−−−−−−−→

^{m Z}

−−−−−−−→

^Z

−−−−−→

⁰

0

−−−→ −−−→ −−−→ −−−→

⁰

0 H (n Sⁿ) H (n Mf) H (n Mf Sⁿ)

m

== == == ,

But this sequence does not split since Z is not isomorphic to Z⊕Z_m if m > 1 , so the retraction cannot exist. In the simplest case of the degree 2 map S¹

→

^{S1, z}

,

^z2,

this says that the M¨obius band does not retract onto its boundary circle.

Homology of Groups

In §1.B we constructed for each group G a CW complex K(G, 1) having a con-tractible universal cover, and we showed that the homotopy type of such a space K(G, 1) is uniquely determined by G . The homology groups H_n K(G, 1)

therefore depend only onG , and are usually denoted simply H_n(G) . The calculations for lens spaces in Example 2.43 show that H_n(Zm) is Zm for odd n and 0 for even n > 0 . Since S¹ is a K(Z, 1) and the torus is a K(Z×Z, 1), we also know the homology of these two groups. More generally, the homology of finitely generated abelian groups can be computed from these examples using the K¨unneth formula in§3.B and the fact that a product K(G, 1)×K(H, 1) is a K(G×H, 1).

Here is an application of the calculation of H_n(Zm) :

在文檔中 Allen Hatcher (頁 146-158)