Homology with Coefficients

The third column maps to 0

→

^Cⁿ^{(Y )}

→

^Cⁿ^(X)

→

^Cⁿ^{(X, Y )}

→

0 , inducing maps of homology groups that are isomorphisms for theX and Y terms as we have seen above.

So by the five-lemma the mapsC_n(A+B, C +D)

→

^Cn(X, Y ) also induce isomorphisms on homology. The relative Mayer–Vietoris sequence is then the long exact sequence of homology groups associated to the short exact sequence of chain complexes given by the third row of the diagram.

L

emma 2.49. If f : S^k

→

^S^k has degree m , then f_∗:H_k(S^k;G)

→

^H^k^(S^k^;G) is multi-plication by m .

P

roof: As a preliminary observation, note that a homomorphism ϕ : G₁

→

^G² ^induces

maps ϕ_]:C_n(X, A; G₁)

→

^Cⁿ^{(X, A; G}²) commuting with boundary maps, so there are induced homomorphisms ϕ_∗:H_n(X, A; G₁)

→

^Hn(X, A; G₂) . These have various nat-urality properties. For example, they give a commutative diagram mapping the long exact sequence of homology for the pair(X, A) with G₁coefficients to the correspond-ing sequence withG₂coefficients. Also, the mapsϕ_∗commute with homomorphisms f_∗ induced by maps f : (X, A)

→

^{(Y , B) .}

Now let f : S^k

→

^S^k have degree m and let ϕ :Z

→

G take 1 to a given element g ∈ G. Then we have a commutative

diagram as at the right, where commu-

−−−−−−→

^f^∗ ^{H (}^k ^S

−−−−−→

⁾

−−−−−→

∗

f_∗

∼ k ≈

−−−−−−→

^{H (}^∼^k ^{S G}^k ⁾ ^≈^G

≈

∗ ϕ ϕ

;

−−−−−→

k S H (∼ k )

k S G H (∼ k )

;

Z Z

−−−−−→

Z tativity of the outer two squares comes from the inductive calculation of these

homology groups, reducing to the casek= 0 when the commutativity is obvious.

Since the diagram commutes, the assumption that the map across the top takes 1 tom implies that the map across the bottom takes g to mg . tu

E

xample 2.50. It is instructive to see what happens to the homology of RPⁿ when the coefficient groupG is chosen to be a field F . The cellular chain complex is

···

---→

⁰ ^F

---→

² ^F

---→

⁰ ^F

---→

² ^F

---→

⁰ ^F

-→

⁰

Hence if F has characteristic 2 , for example if F = Z₂, then H_k(RPⁿ;F ) ≈ F for 0≤ k ≤ n, a more uniform answer than with Z coefficients. On the other hand, if F has characteristic different from 2 then the boundary maps F

---→

² F are isomor-phisms, henceH_k(RPⁿ;F ) is F for k= 0 and for k = n odd, and is zero otherwise.

In§3.A we will see that there is a general algebraic formula expressing homology with arbitrary coefficients in terms of homology withZ coefficients. Some easy special cases that give much of the flavor of the general result are included in the Exercises.

In spite of the fact that homology with Z coefficients determines homology with other coefficient groups, there are many situations where homology with a suitably chosen coefficient group can provide more information than homology with Z coef-ficients. A good example of this is the proof of the Borsuk–Ulam theorem using Z₂ coefficients in§2.B.

As another illustration, we will now give an example of a map f : X

→

^{Y with the}

property that the induced maps f_∗ are trivial for homology with Z coefficients but not for homology with Z_m coefficients for suitably chosen m . Thus homology with Zm coefficients tells us thatf is not homotopic to a constant map, which we would not know using onlyZ coefficients.

E

xample 2.51. Let X be a Moore space M(Zm, n) obtained from Sⁿ by attaching a cell eⁿ⁺¹ by a map of degree m . The quotient map f : X

→

^X/Sⁿ ^{= S}ⁿ⁺¹ ^induces

trivial homomorphisms on reduced homology with Z coefficients since the nonzero reduced homology groups ofX and Sⁿ⁺¹ occur in different dimensions. But withZm

coefficients the story is different, as we can see by considering the long exact sequence of the pair (X, Sⁿ) , which contains the segment

0= eH_n+1(Sⁿ;Zm)

-→

^H^en+1(X;Zm)

---→

^f^∗ ^H^en+1(X/Sⁿ;Zm)

Exactness says that f_∗ is injective, hence nonzero since eH_n+1(X;Z_m) is Z_m, the cel-lular boundary map H_n+1(Xⁿ⁺¹, Xⁿ;Zm)

→

^Hⁿ^(Xⁿ^{, X}ⁿ⁻¹^;^Z^m^{) being}^Z^m

---→

^m ^Z^m^.

Exercises

1. Prove the Brouwer fixed point theorem for maps f : Dⁿ

→

^Dⁿ by applying degree theory to the map Sⁿ

→

^Sⁿ that sends both the northern and southern hemispheres ofSⁿ to the southern hemisphere viaf . [This was Brouwer’s original proof.]

2. Given a map f : S²ⁿ

→

^S²ⁿ, show that there is some point x ∈ S²ⁿ with either f (x) = x or f (x) = −x . Deduce that every map RP²ⁿ

→

^RP²ⁿ has a fixed point.

Construct maps RP²ⁿ⁻¹

→

^RP²ⁿ⁻¹ without fixed points from linear transformations R²ⁿ

→

^R²ⁿ without eigenvectors.

3. Let f : Sⁿ

→

^Sⁿ be a map of degree zero. Show that there exist points x, y ∈ Sⁿ with f (x) = x and f (y) = −y . Use this to show that if F is a continuous vector field defined on the unit ballDⁿ inRⁿsuch that F (x)≠ 0 for all x , then there exists a point on ∂Dⁿ where F points radially outward and another point on ∂Dⁿ where F points radially inward.

4. Construct a surjective map Sⁿ

→

^Sⁿ of degree zero, for each n≥ 1.

5. Show that any two reflections of Sⁿ across different n dimensional hyperplanes are homotopic, in fact homotopic through reflections. [The linear algebra formula for a reflection in terms of inner products may be helpful.]

6. Show that every map Sⁿ

→

^Sⁿ can be homotoped to have a fixed point if n > 0 . 7. For an invertible linear transformation f :Rⁿ

→

^Rⁿ show that the induced map on H_n(Rⁿ,Rⁿ− {0}) ≈ eH_n−1(Rⁿ− {0}) ≈ Z is ¹1 or −¹1 according to whether the determinant of f is positive or negative. [Use Gaussian elimination to show that the matrix of f can be joined by a path of invertible matrices to a diagonal matrix with

±1’s on the diagonal.]

8. A polynomial f (z) with complex coefficients, viewed as a mapC

→

C, can always be extended to a continuous map of one-point compactifications bf : S²

→

^S²^{. Show}

that the degree of bf equals the degree of f as a polynomial. Show also that the local degree of bf at a root of f is the multiplicity of the root.

9. Compute the homology groups of the following 2 complexes:

(a) The quotient of S² obtained by identifying north and south poles to a point.

(b) S¹×(S¹∨ S¹) .

(c) The space obtained fromD²by first deleting the interiors of two disjoint subdisks in the interior of D² and then identifying all three resulting boundary circles together via homeomorphisms preserving clockwise orientations of these circles.

(d) The quotient space ofS¹×S¹obtained by identifying points in the circleS¹×{x0} that differ by 2π /m rotation and identifying points in the circle {x0}×S¹ that differ by 2π /n rotation.

10. Let X be the quotient space of S² under the identificationsx∼ −x for x in the equatorS¹. Compute the homology groupsH_i(X) . Do the same for S³with antipodal points of the equatorial S²⊂ S³ identified.

11. In an exercise for§1.2 we described a 3 dimensional CW complex obtained from the cube I³ by identifying opposite faces via a one-quarter twist. Compute the ho-mology groups of this complex.

12. Show that the quotient map S¹×S¹

→

^S² collapsing the subspace S¹∨ S¹ to a point is not nullhomotopic by showing that it induces an isomorphism onH₂. On the other hand, show via covering spaces that any map S²

→

^S¹^×S¹ is nullhomotopic.

13. Let X be the 2 complex obtained from S¹with its usual cell structure by attaching two 2 cells by maps of degrees 2 and 3 , respectively.

(a) Compute the homology groups of all the subcomplexes A ⊂ X and the corre-sponding quotient complexes X/A .

(b) Show that X ' S², and that the only subcomplex A⊂ X with X/A ' S² is the trivial subcomplex consisting of the 0 cell alone.

14. A map f : Sⁿ

→

^Sⁿ ^satisfying^{f (x)}= f (−x) for all x is called an even map. Show that an even mapSⁿ

→

^Sⁿmust have even degree, and that the degree must in fact be zero when n is even. When n is odd, show there exist even maps of any given even degree. [Hints: If f is even, it factors as a composition Sⁿ

→

^RPⁿ

→

^Sⁿ. Using the calculation of H_n(RPⁿ) in the text, show that the induced map H_n(Sⁿ)

→

^Hⁿ^(RPⁿ⁾

sends a generator to twice a generator when n is odd. It may be helpful to show that the quotient mapRPⁿ

→

^RPⁿ^/RPⁿ⁻¹ induces an isomorphism onH_n whenn is odd.]

15. Show that if X is a CW complex then H_n(Xⁿ) is free by identifying it with the kernel of the cellular boundary mapH_n(Xⁿ, Xⁿ⁻¹)

→

^Hn−1(Xⁿ⁻¹, Xⁿ⁻²) .

16. Let ∆ⁿ = [v₀,··· , v_n] have its natural ∆ complex structure with k simplices [v_i₀,··· , v_i_k] for i₀<··· < i_k. Compute the ranks of the simplicial (or cellular) chain groups∆i(∆ⁿ) and the subgroups of cycles and boundaries. [Hint: Pascal’s triangle.]

Apply this, using also the previous problem, to show that the k skeleton of ∆ⁿ has homology groups eH_i (∆ⁿ)^k

equal to 0 fori < k , and free of rank

n k+1

for i= k.

17. Show the isomorphism between cellular and singular homology is natural in the following sense: A map f : X

→

Y that is cellular — satisfying f (Xⁿ) ⊂ Yⁿ for all n — induces a chain map f_∗ between the cellular chain complexes of X and Y , and the map f_∗:H_n^CW(X)

→

^Hⁿ^CW(Y ) induced by this chain map corresponds to f_∗:H_n(X)

→

^Hⁿ(Y ) under the isomorphism H_n^CW ≈ Hn.

18. For a CW pair (X, A) show there is a relative cellular chain complex formed by the groups H_i(Xⁱ, Xⁱ⁻¹∪ Aⁱ) , having homology groups isomorphic to H_n(X, A) . 19. Compute H_i(RPⁿ/RP^m) for m < n by cellular homology, using the standard CW structure on RPⁿ with RP^m as its m skeleton.

20. For finite CW complexes X and Y , show that χ (X×Y ) = χ (X)χ (Y ).

21. If a finite CW complex X is the union of subcomplexes A and B , show that χ (X)= χ (A) + χ (B) − χ (A ∩ B).

22. For X a finite CW complex and p : eX

→

X an n sheeted covering space, show that χ ( eX)= nχ (X).

23. Show that if the closed orientable surface M_g of genus g is a covering space of M_h, then g = n(h − 1) + 1 for some n, namely, n is the number of sheets in the covering. [Conversely, if g = n(h − 1) + 1 then there is an n sheeted covering M_g

→

^M^h, as we saw in Example 1.41.]

24. Suppose we build S² from a finite collection of polygons by identifying edges in pairs. Show that in the resulting CW structure on

S² the 1 skeleton cannot be either of the two graphs shown, with five and six vertices. [This is one step in a proof that neither of these graphs embeds inR².]

25. Show that for each n∈ Z there is a unique function ϕ assigning an integer to each finite CW complex, such that (a) ϕ(X)= ϕ(Y ) if X and Y are homeomorphic, (b) ϕ(X)= ϕ(A) + ϕ(X/A) if A is a subcomplex of X , and (c) ϕ(S⁰)= n. For such a function ϕ , show that ϕ(X)= ϕ(Y ) if X ' Y .

26. For a pair (X, A) , let X∪ CA be X with a cone on A attached.

(a) Show thatX is a retract of X∪CA iff A is contractible in X : There is a homotopy f_t:A

→

^{X with f}⁰ the inclusionA

>

^{X and f}¹ a constant map.

(b) Show that if A is contractible in X then H_n(X, A)≈ eH_n(X)⊕H^e_n−1(A) , using the fact that(X∪ CA)/X is the suspension SA of A.

27. The short exact sequences 0

→

^Cn(A)

→

^Cn(X)

→

^Cn(X, A)

→

0 always split, but why does this not always yield splittingsH_n(X)≈ Hn(A)⊕H_n(X, A) ?

28. (a) Use the Mayer–Vietoris sequence to compute the homology groups of the space obtained from a torusS¹×S¹by attaching a M¨obius band via a homeomorphism from the boundary circle of the M¨obius band to the circle S¹×{x₀} in the torus.

(b) Do the same for the space obtained by attaching a M¨obius band to RP² via a homeomorphism of its boundary circle to the standard RP¹⊂ RP².

29. The surface M_g of genus g , embedded in R³ in the standard way, bounds a compact regionR . Two copies of R , glued together by the identity map between their boundary surfaces M_g, form a closed 3-manifoldX . Compute the homology groups ofX via the Mayer–Vietoris sequence for this decomposition of X into two copies of R . Also compute the relative groups H_i(R, M_g) .

30. For the mapping torus T_f of a mapf : X

→

X , we constructed in Example 2.48 a long exact sequence ···

-→

^Hⁿ^(X)

---→

^1−f^∗ ^Hⁿ^(X)

-→

^Hⁿ^(T^f⁾

-→

^Hn−1(X)

-→

^{···. Use}

this to compute the homology of the mapping tori of the following maps:

(a) A reflection S²

→

^S²^.

(b) A map S²

→

^S² of degree 2 .

→

^S¹^×S¹ that is the identity on one factor and a reflection on the other.

(d) The mapS¹×S¹

→

^S¹^×S¹ that is a reflection on each factor.

(e) The mapS¹×S¹

→

^S¹^×S¹ that interchanges the two factors and then reflects one of the factors.

31. Use the Mayer–Vietoris sequence to show there are isomorphisms eH_n(X∨ Y ) ≈ He_n(X)⊕H^e_n(Y ) if the basepoints of X and Y that are identified in X∨ Y are defor-mation retracts of neighborhoods U⊂ X and V ⊂ Y .

32. For SX the suspension of X , show by a Mayer–Vietoris sequence that there are isomorphisms eH_n(SX)≈ eH_n−1(X) for all n .

33. Suppose the space X is the union of open sets A₁,··· , A_n such that each inter-sectionA_i₁∩ ··· ∩ Ai_k is either empty or has trivial reduced homology groups. Show that eH_i(X) = 0 for i ≥ n − 1, and give an example showing this inequality is best possible, for each n .

34. Derive the long exact sequence of a pair (X, A) from the Mayer–Vietoris sequence applied to X ∪ CA, where CA is the cone on A. [We showed after the proof of Proposition 2.22 thatH_n(X, A)≈ eH_n(X∪ CA) for all n.]

35. Use the Mayer–Vietoris sequence to show that a nonorientable closed surface, or more generally a finite simplicial complex X for which H₁(X) contains torsion, cannot be embedded as a subspace of R³ in such a way as to have a neighborhood homeomorphic to the mapping cylinder of some map from a closed orientable surface to X . [This assumption on a neighborhood is in fact not needed if one deduces the result from Alexander duality in§3.3.]

36. Show that H_i(X×Sⁿ) ≈ Hi(X)⊕H_i−n(X) for all i and n , where H_i = 0 for i < 0 by definition. Namely, show H_i(X×Sⁿ) ≈ Hi(X)⊕H_i(X×Sⁿ, X×{x0}) and H_i(X×Sⁿ, X×{x0}) ≈ H_i−1(X×Sⁿ⁻¹, X×{x0}). [For the latter isomorphism the rela-tive Mayer–Vietoris sequence yields an easy proof.]

37. Give an elementary derivation for the Mayer–Vietoris sequence in simplicial ho-mology for a ∆ complex X decomposed as the union of subcomplexes A and B .

38. Show that a commutative diagram

A A

−−−−→

−−−−→ −−−−→

−−−→

−−−−→ −−−−→ −−−−→ −−−−→

−−−−→

−−−→

−−−−→

−−−→

n 1+

-n 1

-. -. -.

n 1

. . .

. . . . . .

with the two sequences across the top and bottom exact, gives rise to an exact se-quence ···

-→

^En+1

-→

^Bⁿ

-→

^Cⁿ^⊕^Dⁿ

-→

^Eⁿ

-→

^Bn−1

-→

··· where the maps are obtained from those in the previous diagram in the obvious way, except that B_n

→

^Cⁿ^⊕^Dⁿ has a minus sign in one coordinate.

39. Use the preceding exercise to derive relative Mayer–Vietoris sequences for CW pairs(X, Y )= (A ∪ B, C ∪ D) with A = B or C = D .

40. From the long exact sequence of homology groups associated to the short ex-act sequence of chain complexes 0

-→

^Cⁱ^(X)

---→

ⁿ ^Cⁱ^(X)

-→

^Cⁱ^(X;^Zⁿ⁾

-→

^{0 deduce}

immediately that there are short exact sequences

-→

^Hⁱ^(X)/nHⁱ^(X)

-→

^Hⁱ^(X;^Zⁿ⁾

-→

n-Torsion(H_i−1(X))

-→

⁰

where n-Torsion(G) is the kernel of the map G

---→

ⁿ ^{G , g}

,

ng . Use this to show that He_i(X;Zp)= 0 for all i and all primes p iff eH_i(X) is a vector space over Q for all i.

41. For X a finite CW complex and F a field, show that the Euler characteristic χ (X) can also be computed by the formula χ(X)=P

n(−1)ⁿdimH_n(X; F ) , the alternating sum of the dimensions of the vector spaces H_n(X; F ) .

42. Let X be a finite connected graph having no vertex that is the endpoint of just one edge, and suppose that H₁(X;Z) is free abelian of rank n > 1, so the group of automorphisms of H₁(X;Z) is GL_n(Z), the group of invertible n×n matrices with integer entries whose inverse matrix also has integer entries. Show that ifG is a finite group of homeomorphisms of X , then the homomorphism G

→

^GLⁿ(Z) assigning to g : X

→

X the induced homomorphism g_∗:H₁(X;Z)

→

^H¹^(X;Z) is injective. Show the same result holds if the coefficient groupZ is replaced by Zmwith m > 2 . What goes wrong when m= 2?

43. (a) Show that a chain complex of free abelian groups C_n splits as a direct sum of subcomplexes 0

→

^Ln+1

→

^Kn

→

0 with at most two nonzero terms. [Show the short exact sequence 0

→

^Ker^∂

→

^Cⁿ

→

^Im^∂

→

0 splits and takeK_n= Ker ∂ .]

(b) In case the groups C_n are finitely generated, show there is a further splitting into summands 0

→

^{0 and 0}

-→

---→

^m ^Z

-→

0 . [Reduce the matrix of the boundary map L_n+1

→

^Kⁿ to echelon form by elementary row and column operations.]

(c) Deduce that if X is a CW complex with finitely many cells in each dimension, then H_n(X; G) is the direct sum of the following groups:

a copy ofG for eachZ summand of Hn(X) a copy ofG/mG for each Zm summand of H_n(X)

a copy of the kernel ofG

---→

^m ^{G for each}^Z^m ^{summand of} ^Hn−1(X)

Sometimes it is good to step back from the forest of details and look for gen-eral patterns. In this rather brief section we will first describe the gengen-eral pattern of homology by axioms, then we will look at some common formal features shared by many of the constructions we have made, using the language of categories and functors which has become common in much of modern mathematics.

在文檔中 Allen Hatcher (頁 162-169)