Analyze what happens when the number of handles in the basic building block for the Alexander horned sphere is doubled, as in

the figure at the right.

8. Show that R²ⁿ⁺¹ is not a division algebra over R if n > 0 by showing that if it were, then for nonzero a∈ R²ⁿ⁺¹ the map S²ⁿ

→

^S2n,x

,

ax/|ax| would be homotopic to x

,

^−ax/|ax|,

but these maps have different degrees.

9. Make the transfer sequence explicit in the case of a trivial covering eX

→

^{X , where}

Xe= X×S⁰.

10. Use the transfer sequence for the covering S^∞

→

^{RP∞ to computeHn(RP∞;Z2) .}

11. Use the transfer sequence for the covering X×S^∞

→

^X×RP∞ to produce isomor-phismsH_n(X×RP^∞;Z₂)≈L

i≤nH_i(X;Z₂) for all n .

Many spaces of interest in algebraic topology can be given the structure of sim-plicial complexes, and early in the history of the subject this structure was exploited as one of the main technical tools. Later, CW complexes largely superseded simplicial complexes in this role, but there are still some occasions when the extra structure of simplicial complexes can be quite useful. This will be illustrated nicely by the proof of the classical Lefschetz fixed point theorem in this section.

One of the good features of simplicial complexes is that arbitrary continuous maps between them can always be deformed to maps that are linear on the simplices of some subdivision of the domain complex. This is the idea of ‘simplicial approxi-mation,’ developed by Brouwer and Alexander before 1920. Here is the relevant def-inition: If K and L are simplicial complexes, then a map f : K

→

L is simplicial if it sends each simplex of K to a simplex of L by a linear map taking vertices to ver-tices. In barycentric coordinates, a linear map of a simplex[v₀,··· , v_n] has the form P

it_iv_i

,

^{Pitif (vi}) . Since a linear map from a simplex to a simplex is uniquely determined by its values on vertices, this means that a simplicial map is uniquely determined by its values on vertices. It is easy to see that a map from the vertices of K to the vertices of L extends to a simplicial map iff it sends the vertices of each simplex ofK to the vertices of some simplex of L .

Here is the most basic form of theSimplicial Approximation Theorem:

T

heorem 2C.1. If K is a finite simplicial complex and L is an arbitrary simplicial complex, then any map f : K

→

L is homotopic to a map that is simplicial with respect to some iterated barycentric subdivision of K .

To see that subdivision ofK is essential, consider the case of maps Sⁿ

→

^{Sn. With}

fixed simplicial structures on the domain and range spheres there are only finitely many simplicial maps since there are only finitely many ways to map vertices to ver-tices. Hence only finitely many degrees are realized by maps that are simplicial with respect to fixed simplicial structures in both the domain and range spheres. This re-mains true even if the simplicial structure on the range sphere is allowed to vary, since if the range sphere has more vertices than the domain sphere then the map cannot be surjective, hence must have degree zero.

Before proving the simplicial approximation theorem we need some terminology and a lemma. Thestar St σ of a simplex σ in a simplicial complex X is defined to be the subcomplex consisting of all the simplices ofX that contain σ . Closely related to this is theopen star st σ , which is the union of the interiors of all simplices containing σ , where the interior of a simplex τ is by definition τ− ∂τ . Thus st σ is an open set in X whose closure is St σ .

L

emma 2C.2. For vertices v₁,··· , v_n of a simplicial complex X , the intersection stv₁∩ ··· ∩ st v_n is empty unless v₁,··· , v_n are the vertices of a simplex σ of X , in which case st v₁∩ ··· ∩ st vn= st σ .

P

roof: The intersection st v₁∩ ··· ∩ st v_n consists of the interiors of all simplicesτ whose vertex set contains {v1,··· , vn}. If st v1∩ ··· ∩ st vn is nonempty, such a τ exists and contains the simplex σ = [v1,··· , vn] ⊂ X . The simplices τ containing {v1,··· , vn} are just the simplices containing σ , so st v1∩ ··· ∩ st vn= st σ . tu

P

roof of 2C.1: Choose a metric on K that restricts to the standard Euclidean metric on each simplex of K . For example, K can be viewed as a subcomplex of a sim-plex ∆^N whose vertices are all the vertices of K , and we can restrict a standard met-ric on ∆^N to give a metric on K . Let ε be a Lebesgue number for the open cover { f⁻¹ stw

| w is a vertex of L } of K . After iterated barycentric subdivision of K we may assume that each simplex has diameter less than ε/2 . The closed star of each vertex v of K then has diameter less than ε , hence this closed star maps by f to the open star of some vertex g(v) of L . The resulting map g : K⁰

→

^L0 thus satisfies f (St v)⊂ st g(v) for all vertices v of K .

To see that g extends to a simplicial map g : K

→

L , consider the problem of extending g over a simplex [v₁,··· , v_n] of K . An interior point x of this simplex lies in stv_i for eachi , so f (x) lies in st g(v_i) for each i , since f (st v_i)⊂ st g(v_i) by the definition ofg(v_i) . Thus st g(v₁)∩···∩st g(vn)≠ ∅, so [g(v1),··· , g(vn)] is a simplex ofL by the lemma, and we can extend g linearly over [v₁,··· , vn] . Both f (x) andg(x) lie in a single simplex of L since g(x) lies in [g(v₁),··· , g(v_n)] and f (x) lies in the star of this simplex. So taking the linear path(1−t)f (x)+tg(x), 0 ≤ t ≤ 1, in the simplex containingf (x) and g(x) defines a homotopy from f to g . To check continuity of this homotopy it suffices to restrict to the simplex [v₁,··· , vn] , where

continuity is clear since f (x) varies continuously in the star of [g(v₁),··· , g(vn)]

and g(x) varies continuously in [g(v₁),··· , g(v_n)] . tu Notice that if f already sends some vertices of K to vertices of L then we may chooseg to equal to f on these vertices, and hence the homotopy from f to g will be stationary on these vertices. This is convenient if one is in a situation where one wants maps and homotopies to preserve basepoints.

The proof makes it clear that the simplicial approximation g can be chosen not just homotopic to f but also close to f if we allow subdivisions of L as well as K . The Lefschetz Fixed Point Theorem

This very classical application of homology is a considerable generalization of the Brouwer fixed point theorem. It is also related to the Euler characteristic formula.

For a homomorphism ϕ :Zⁿ

→

^Zn with matrix [a_ij] , the trace tr ϕ is defined to be P

ia_ii, the sum of the diagonal elements of [a_ij] . Since tr([a_ij][b_ij]) = tr([b_ij][a_ij]) , conjugate matrices have the same trace, and it follows that tr ϕ is in-dependent of the choice of basis forZⁿ. For a homomorphism ϕ : A

→

A of a finitely generated abelian group A we can then define tr ϕ to be the trace of the induced homomorphismϕ : A/Torsion

→

A/Torsion .

For a mapf : X

→

X of a finite CW complex X , or more generally any space whose homology groups are finitely generated and vanish in high dimensions, theLefschetz number τ(f ) is defined to be P

n(−1)ⁿtr f_∗:H_n(X)

→

^Hn(X)
. In particular, if f is the identity, or is homotopic to the identity, then τ(f ) is the Euler characteristic χ (X) since the trace of the n×n identity matrix is n.

Here is the Lefschetz fixed point theorem:

T

heorem 2C.3. If X is a finite simplicial complex, or more generally a retract of a finite simplicial complex, and f : X

→

X is a map with τ(f )≠ 0, then f has a fixed point.

As we show in Theorem A.7 in the Appendix, every compact, locally contractible space that can be embedded in Rⁿ for some n is a retract of a finite simplicial com-plex. This includes compact manifolds and finite CW complexes, for example. The compactness hypothesis is essential, since a translation of R has τ = 1 but no fixed points. For an example showing that local properties are also

significant, let X be the compact subspace of R² consisting of two concentric circles together with a copy of R between them whose two ends spiral in to the two circles, wrapping around them infinitely often, and let f : X

→

X be a homeomorphism translating the copy of R along itself and rotating the circles,

with no fixed points. Since f is homotopic to the identity, we have τ(f ) = χ (X), which equals 1 since the three path components ofX are two circles and a line.

If X has the same homology groups as a point, at least modulo torsion, then the theorem says that every map X

→

X has a fixed point. This holds for exam-ple for RPⁿ if n is even. The case of projective spaces is interesting because of its connection with linear algebra. An invertible linear transformation f :Rⁿ

→

^Rn

takes lines through 0 to lines through 0 , hence induces a map f :RPⁿ⁻¹

→

^RPn−1.

Fixed points of f are equivalent to eigenvectors of f . The characteristic polyno-mial of f has odd degree if n is odd, hence has a real root, so an eigenvector ex-ists in this case. This is in agreement with the observation above that every map RP^2k

→

^RP2k has a fixed point. On the other hand the rotation of R^2k defined by f (x₁,··· , x2k)= (x2,−x1, x₄,−x3,··· , x2k,−x_2k−1) has no eigenvectors and its pro-jectivization f :RP^2k−1

→

^RP2k−1 has no fixed points.

Similarly, in the complex case an invertible linear transformation f :Cⁿ

→

^Cn

in-duces f :CPⁿ⁻¹

→

^CPn−1, and this always has a fixed point since the characteristic polynomial always has a complex root. Nevertheless, as in the real case there is a map CP^2k−1

→

^CP2k−1 without fixed points. Namely, consider f :C^2k

→

^{C2k defined}

by f (z₁,··· , z2k) = (z2,−z1, z₄,−z3,··· , z2k,−z_2k−1) . This map is only ‘conjugate-linear’ over C, but this is still good enough to imply that f induces a well-defined map f on CP^2k−1, and it is an easy algebra exercise to check that f has no fixed points. The similarity between the real and complex cases persists in the fact that every map CP^2k

→

^CP2k has a fixed point, though to deduce this from the Lefschetz fixed point theorem one needs more structure than just homology, so this will be left as an exercise for§3.2, using cup products.

One could go further and consider the quaternionic case. Oddly enough, every map HPⁿ

→

^HPn has a fixed point if n > 1 , according to an exercise in§4.K. When n= 1 the antipodal map of S³= HP¹ has no fixed points.

P

roof of 2C.3: The general case easily reduces to the case of finite simplicial com-plexes, for suppose r : K

→

X is a retraction of the finite simplicial complex K onto X . For a map f : X

→

X , the composition f r : K

→

^X ⊂ K then has exactly the same fixed points asf . Since r_∗:H_n(K)

→

^Hn(X) is projection onto a direct summand, we clearly have tr(f_∗r_∗)= tr f_∗, soτ(f_∗r_∗)= τ(f_∗) .

ForX a finite simplicial complex, suppose that f : X

→

X has no fixed points. We claim there is a subdivisionL of X , a further subdivision K of L , and a simplicial map g : K

→

L homotopic to f such that g(σ )∩σ = ∅ for each simplex σ of K . To see this, first choose a metricd on X as in the proof of the simplicial approximation theorem.

Since f has no fixed points, d x, f (x)

> 0 for all x∈ X , so by the compactness of X there is an ε > 0 such that d x, f (x)

> ε for all x . Choose a subdivision L of X so that the stars of all simplices have diameter less thanε/2 . Applying the simplicial approximation theorem, there is a subdivision K of L and a simplicial map g : K

→

^L

homotopic tof . By construction, this g has the property that for each simplex σ of K , f (σ ) is contained in the star of the simplex g(σ ) . We may assume the subdivision

K is chosen fine enough so that its simplices all have diameter less than ε/2 . Then g(σ )∩ σ = ∅ for each simplex σ of K since for x ∈ σ , σ lies within distance ε/2 ofx and g(σ ) lies within distance ε/2 of f (x) , while d x, f (x)

> ε .

For such a g : K

→

L , the Lefschetz numbers τ(f ) and τ(g) are equal since f and g are homotopic. Since g is simplicial, it takes the n skeleton Kⁿ of K to the n skeleton Lⁿ ofL , for each n . Since K is a subdivision of L , Lⁿis contained inKⁿ, and hence g(Kⁿ)⊂ Kⁿ for all n . Thus g induces a chain map of the cellular chain complex {Hn(Kⁿ, Kⁿ⁻¹)} to itself. This can be used to compute τ(g) according to the formula

τ(g)=X

n

(−1)ⁿtr g_∗:H_n(Kⁿ, Kⁿ⁻¹)

→

^Hn(Kⁿ, Kⁿ⁻¹)

This is the analog of Theorem 2.44 for trace instead of rank, and is proved in precisely the same way, based on the elementary algebraic fact that trace is additive for endo-morphisms of short exact sequences: Given a

com-mutative diagram as at the right with exact rows, 0 A

α β γ

−−−→ −−−→ −−−→

−−→ −−−→

^B

−−−→

^C

−−→

⁰

0

−−→

A

−−−→

^B

−−−→

^C

−−→

⁰

then trβ = tr α + tr γ . This algebraic fact can be proved by reducing to the easy case that A , B , and

C are free by first factoring out the torsion in B , hence also in A , then eliminating any remaining torsion in C by replacing A by a larger subgroup A⁰ ⊂ B , with A having finite index in A⁰. The details of this argument are left to the reader.

Finally, note that g_∗:H_n(Kⁿ, Kⁿ⁻¹)

→

^{Hn(Kn, Kn−1}) has trace 0 since the matrix for g_∗ has zeros down the diagonal, in view of the fact that g(σ )∩ σ = ∅ for each

n simplex σ . So τ(f )= τ(g) = 0. tu

E

xample 2C.4. Let us verify the theorem in an example. Let X be the closed ori-entable surface of genus 3 as shown in the figure below, with f : X

→

^{X the 180}

degree rotation about a vertical axis passing through the central hole of X . Since f has no fixed points, we

1 1

α β

2 2

α α2

β

3 3

α β 0

should have τ(f )= 0. The induced map f_∗:H₀(X)

→

^H0(X) is the

iden-tity, as always for a path-connected space, so this contributes 1 to τ(f ) . For H₁(X) we saw in Example 2A.2 that the six loops α_i and β_i represent a basis. The map f_∗ interchanges the homology classes of α₁ and α₃, and likewise for β₁ and β₃, while β₂ is sent to itself and α₂ is sent to α⁰₂ which is homologous to α₂ as we saw in Example 2A.2. So f_∗:H₁(X)

→

^H1(X) contributes −2 to τ(f ). It remains to check that f_∗:H₂(X)

→

^H2(X) is the

iden-tity, which we do by the commutative diagram at the right, wherex is a point of X in the

cen-−−−−−→ −−−−−−−−−−−→

− − − − − →

X H2( )

X x X

H2( , -{ }) ^f∗ H2(X,X-{y})

f∗

≈ ^H2

−−−−−→

⁽_≈^X)

tral torus and y = f (x). We can see that the

left-hand vertical map is an isomorphism by considering the long exact sequence of the triple (X, X− {x}, X¹) where X¹ is the 1 skeleton of X in its usual CW struc-ture and x is chosen in X− X¹, so that X− {x} deformation retracts onto X¹ and H_n(X− {x}, X¹) = 0 for all n. The same reasoning shows the right-hand vertical map is an isomorphism. There is a similar commutative diagram with f replaced by a homeomorphism g that is homotopic to the identity and equals f in a neighbor-hood of x , with g the identity outside a disk in X containing x and y . Since g is homotopic to the identity, it induces the identity across the top row of the diagram, and since g equals f near x , it induces the same map as f in the bottom row of the diagram, by excision. It follows that the map f_∗ in the upper row is the identity.

This example generalizes to surfaces of any odd genus by adding symmetric pairs of tori at the left and right. Examples for even genus are described in one of the exercises.

Fixed point theory is a well-developed side branch of algebraic topology, but we touch upon it only occasionally in this book. For a nice introduction see [Brown 1971].

Simplicial Approximations to CW Complexes

The simplicial approximation theorem allows arbitrary continuous maps to be replaced by homotopic simplicial maps in many situations, and one might wonder about the analogous question for spaces: Which spaces are homotopy equivalent to simplicial complexes ? We will show this is true for the most common class of spaces in algebraic topology, CW complexes. In the Appendix the question is answered for a few other classes of spaces as well.

T

heorem 2C.5. Every CW complex X is homotopy equivalent to a simplicial complex, which can be chosen to be of the same dimension as X , finite if X is finite, and countable if X is countable.

We will build a simplicial complex Y ' X inductively as an increasing union of subcomplexes Y_n homotopy equivalent to the skeleta Xⁿ. For the inductive step, assuming we have already constructed Y_n ' Xⁿ, let eⁿ⁺¹ be an (n+ 1) cell of X attached by a map ϕ : Sⁿ

→

^{Xn. The map Sn}

→

^Yn corresponding to ϕ under the homotopy equivalence Y_n' Xⁿ is homotopic to a simplicial map f : Sⁿ

→

^{Yn by the}

simplicial approximation theorem, and it is not hard to see that the spacesXⁿ∪_ϕeⁿ⁺¹ andY_n∪_feⁿ⁺¹ are homotopy equivalent, where the subscripts denote attachingeⁿ⁺¹ via ϕ and f , respectively; see Proposition 0.18 for a proof. We can view Y_n∪feⁿ⁺¹ as the mapping cone C_f, obtained from the mapping cylinder of f by collapsing the domain end to a point. If we knew that the mapping cone of a simplicial map was a simplicial complex, then by performing the same construction for all the(n+1) cells of X we would have completed the induction step. Unfortunately, and somewhat surprisingly, mapping cones and mapping cylinders are rather awkward objects in the

simplicial category. To avoid this awkwardness we will instead construct simplicial analogs of mapping cones and cylinders that have all the essential features of actual mapping cones and cylinders.

Let us first construct the simplicial analog of a mapping cylinder. For a simpli-cial map f : K

→

L this will be a simplicial complex M(f ) containing both L and the barycentric subdivision K⁰ of K as subcomplexes, and such that there is a defor-mation retractionr_t of M(f ) onto L with r₁||K⁰= f .

The figure shows the case thatf is a simplicial surjec-tion∆²

→

^∆1. The construction proceeds one simplex of K at a time, by induction on dimension. To begin, the ordinary mapping cylinder of f ||K⁰ suffices for M(f ||K⁰) . Assume inductively that we have already constructed M(f ||Kⁿ⁻¹) . Let σ be an n simplex of

K and let τ = f (σ ), a simplex of L of dimension n or less. By the inductive hy-pothesis we have already constructedM(f : ∂σ

→

τ) with the desired properties, and we letM(f : σ

→

τ) be the cone on M(f : ∂σ

→

τ) , as shown in the figure. The space M(f : ∂σ

→

τ) is contractible since by induction it deformation retracts onto τ which is contractible. The cone M(f : σ

→

τ) is of course contractible, so the inclusion of M(f : ∂σ

→

τ) into M(f : σ

→

τ) is a homotopy equivalence. This implies that M(f : σ

→

τ) deformation retracts onto M(f : ∂σ

→

τ) by Corollary 0.20, or one can give a direct argument using the fact that M(f : ∂σ

→

τ) is contractible. Performing these constructions for all n simplices σ , we obtain M(f ||Kⁿ) with a deformation retraction onto M(f ||Kⁿ⁻¹) . Taking the union over all n yields M(f ) with a defor-mation retraction r_t onto L , namely the infinite concatenation of the previous de-formation retractions, with the dede-formation retraction ofM(f ||Kⁿ) onto M(f ||Kⁿ⁻¹) performed in thet interval [1/2ⁿ⁺¹, 1/2ⁿ] . The map r₁||K produced by this process may not equal f , but it is homotopic to f via the linear homotopy tf+ (1 − t)r₁, which is defined sincer₁(σ )⊂ f (σ ) for all simplices σ of K . By applying the homo-topy extension property to the homohomo-topy ofr₁ that equals tf+ (1 − t)r1 on K and the identity map on L , we can improve our deformation retraction of M(f ) onto L so that its restriction to K at time 1 is f .

From the simplicial analog M(f ) of a mapping cylinder we construct the simpli-cial ‘mapping cone’ C(f ) by attaching the ordinary cone on K⁰ to the subcomplex K⁰⊂ M(f ).

P

roof of 2C.5: We will construct for each n a CW complex Z_n containing Xⁿ as a deformation retract and also containing as a deformation retract a subcomplex Y_n that is a simplicial complex. Beginning with Y₀= Z₀ = X⁰, suppose inductively that we have already constructed Y_nandZ_n. Let the cells eⁿ⁺¹_α ofX be attached by maps ϕ_α:Sⁿ

→

^Xn. Using the simplicial approximation theorem, there is a homotopy from ϕ_α to a simplicial map f_α:Sⁿ

→

^Yn. The CW complex W_n= Zn

S

αM(f_α) contains a

simplicial subcomplexS_αⁿhomeomorphic toSⁿat one end ofM(f_α) , and the homeo-morphismSⁿ≈ S_αⁿ is homotopic inW_nto the mapf_α, hence also toϕ_α. LetZ_n+1 be obtained fromZ_nby attachingD_αⁿ⁺¹×I ’s via these homotopies between the ϕ_α’s and the inclusionsS_αⁿ

>

^{Wn. ThusZn+1 containsXn+1}at one end, and at the other end we have a simplicial complex Y_n+1= Yn

S

αC(f_α) , where C(f_α) is obtained from M(f_α) by attaching a cone on the subcomplexS_αⁿ. Since Dⁿ⁺¹×I deformation retracts onto

∂Dⁿ⁺¹×I ∪ Dⁿ⁺¹×{1}, we see that Z_n+1 deformation retracts ontoZ_n∪ Y_n+1, which in turn deformation retracts onto Y_n∪ Y_n+1 = Y_n+1 by induction. Likewise, Z_n+1 deformation retracts ontoXⁿ⁺¹∪Wn which deformation retracts ontoXⁿ⁺¹∪Znand hence onto Xⁿ⁺¹∪ Xⁿ= Xⁿ⁺¹ by induction.

Let Y = S

nY_n and Z = S

nZ_n. The deformation retractions of Z_n onto Xⁿ give deformation retractions ofX∪ Z_n ontoX , and the infinite concatenation of the latter deformation retractions is a deformation retraction of Z onto X . Similarly, Z

deformation retracts onto Y . tu

Exercises

1. What is the minimum number of edges in simplicial complex structures K and L

在文檔中 Allen Hatcher (頁 186-193)