text stringlengths 270 6.81k |
|---|
properties, we will use the transfer in the construction of a number of spaces whose Zp cohomology is a polynomial ring. Let π : X→X be an n sheeted covering space, for some finite n. to the induced map on singular chains π♯ : Ck( phism in the opposite direction τ : Ck(X)→Ck( k→X the sum of the n distinct lifts σ : σ :... |
k(X; F )→H k( the subgroup H ∗( X→X be an n sheeted covering space defined by an acX. Then with coefficients in a field F whose characteristic is 0 consisting of classes α such that γ∗(α) = α for all γ ∈ X; F ) is injective with image e X; F ) on Γ e Γ e. Γ 322 Chapter 3 Cohomology X; F ) Γ e Γ k→ X to the sum of all its ... |
)Zn is all of H 0 and H 1, plus a copy of G in dimension variant cohomology H ∗( k, the cellular cohomology classes assigning the same element of G to each S k. Thus X; G)Zn is exactly the image of π ∗ for i = 0 and k, while the image of π ∗ in H i( X; G)Zn or not dedimension 1 is the subgroup nH 1( pends on G. For G =... |
Y / H ∗(Y / that are invariant under the induced action of, ; Zp) is isomorphic to the subring of Zp[y1, ···, yn] consisting of polynomials Γ on H ∗(Y ; Zp). And in some cases. Then by Proposition 3G.1 above, if p does not divide the order of Γ Γ Γ this subring is itself a polynomial ring. Γ Transfer Homomorphisms Sect... |
: Bπ→Bπ ′, Bϕ([g1| ··· |gn]) = [ϕ(g1)| ··· |ϕ(gn)], satisfying the functor properties B(ϕψ) = BϕBψ and B11 = 11. In particular, if is a group of automorphisms ∆ Γ of π, then acts on Bπ. Γ action of a group space Y ′. Namely, take Y ′ = Y × E where Γ acts on E The other ingredient we shall need is the Borel constructio... |
we have taken K(Zp, 1) = BZp × E and γ takes an edge loop [g] in BZp to [γ(g)] = [mg]. Hence γ acts on H1(K(Zp, 1); Z) by multiplication by m. It follows that γ(α) = mα and γ(β) = mβ since H 1(K(Zp, 1); Zp) ≈ Hom(H1(K(Zp, 1)), Zp) and H 2(K(Zp, 1); Zp) ≈ Ext(H1(K(Zp, 1)), Zp), and it is a gen- Γ Γ Γ 324 Chapter 3 Coho... |
denoting dimension. For a given choice of d the condition that d divides p − 1 Zp Γ says p ≡ 1 mod d, which is satisfied by infinitely many p ’s, according to a classical Λ theorem of Dirichlet. e Example 3G.5. The two preceding examples can be modified so as to eliminate the exterior algebra factors, by replacing Zp by ... |
sequence of subgroups Zpi, the space BZp∞ is the union of the increasing sequence of subcomplexes BZpi. We can therefore apply Transfer Homomorphisms Section 3.G 325 Proposition 3F.5 to conclude that H ∗(K(Zp∞, 1); Zp) is zero in odd dimensions, while in even dimensions the map H ∗(K(Zp∞, 1); Zp)→H ∗(K(Zp, 1); Zp) ind... |
� inducing an isomorphism of H ∗(BZp∞ × H ∗(BZp × p − 1, then taking cohomology the polynomial ring Zp[y2d]. ֓ BZp Example 3G.6. Now we enlarge the preceding example by taking products and bringing in the permutation group to produce a space with Zp cohomology the polynomial ring Zp[y2d, y4d, ···, y2nd] where d is any ... |
entries is 1. This has the effect of enlarging the ring of polynomials invariant under the action, and it can be shown that the invariant Γ 326 Chapter 3 Cohomology polynomials form a polynomial ring Zp[y2d, y4d, ···, y2(n−1)d, y2nq], with the last q i. For example, if n = 2 and q = 1 we obtain generator y2nd replaced ... |
listed in the following table. Lie group S 1 SU(n) Sp(n) SO(2k) G2 F4 E6 E7 E8 degrees 1 2, 3, ···, n 2, 4, ···, 2n 2, 4, ···, 2k − 2, k 2, 6 2, 6, 8, 12 2, 5, 6, 8, 9, 12 2, 6, 8, 10, 12, 14 2, 8, 12, 14, 18, 20, 24, 30 primes all all all The remaining examples form two infinite families plus 30 sporadic exceptions sh... |
1, 3 mod 8 p ≡ 1 mod 8 p ≡ 1, 19 mod 24 p ≡ 1 mod 24 p ≡ 1 mod 5 p ≡ 1 mod 20 p ≡ 1 mod 15 For the prime 2 the realization problem has not yet been completely solved. Among the known examples are those in the table at the right. The construction for the last entry, which does not arise from a Lie group, is in [Dwyer &... |
�cients to nonorientable manifolds is to use local coefficients. In the overall scheme of algebraic topology, however, the role played by local coefficients is fairly small. Local coefficients bring an extra level of complication that one tries to avoid whenever possible. With this in mind, the goal of this section will not ... |
If M is an arbitrary module over Z[π ], we would like to define Cn(X; M) to be X) ⊗ Z[π ]M, but for tensor products over a noncommutative ring one has to be a Cn( little careful with left and right module structures. In general, if R is a ring, possibly noncommutative, one defines the tensor product A ⊗R B of a right R ... |
��nition homology groups with local coefficients. e For cohomology one can set C n(X; M) = HomZ[π ](Cn( homomorphisms Cn( cohomology groups H n(X; M) are cohomology groups with local coefficients. X), M), the Z[π ] module X)→M. These groups C n(X; M) form a cochain complex whose e e Local Coefficients Section 3.H 329 → n→ σ ... |
this, let X ′→X be the cover corresponding to a subgroup π ′ ⊂ π. Then the free abelian group Z[π /π ′] with basis the cosets γπ ′ is a Z[π ] module and X) ⊗Z[π ] Z[π /π ′] ≈ Cn(X ′), so Hn(X; Z[π /π ′]) ≈ Hn(X ′). More generally, if A is Cn( an abelian group then A[π /π ′] is a Z[π ] module and Hn(X; A[π /π ′]) ≈ Hn(... |
ed covering space. If τ is the nontrivial deck transformation of X ′, let n (X ′) = {α ∈ Cn(X ′) | τ♯(α) = −α}. It C + n (X ′) = {α ∈ Cn(X ′) | τ♯(α) = α} and C − n→X ′, and we have follows easily that C ± short exact sequences n (X ′) has basis the chains σ ± τσ for σ : 0 -→ C − 0 -→ C + n (X ′) -→ 0 n (X ′) ֓ Cn(X ′)... |
a Z[π1M] module by letting a loop γ in M act on Z by multiplication by +1 or −1 according to whether γ preserves or reverses local orientations of M. The double cover X ′→X is then the 2 sheeted cover M orientable. The nonorientability of M implies that Hn(M) = 0. Since Hn+1(M) = 0, the exact seM) ≈ Z. This can be int... |
homology coefficient group Z, though one could equally well define a bundle of groups MG→M for any abelian coefficient group G. Homology groups of X with coefficients in a bundle E of abelian groups may n→X is a sinbe defined as follows. Consider finite sums n→E is a lifting of σi. The sum of two lifts ni gular n simplex in X ... |
isomorphic to G. However if E is not a product, there is no canonical isomorphism between different fibers p−1(x), so one cannot identify Hn(X; E) with ordinary homology. In the general case, lifts ni : ∆ ∆ An alternative approach would be to take the coefficients ni to be elements of n), say σi(v0). However, with such a ... |
paths between two points, and loops give rise to an action of π1 on the fibers. In the case of bundles of groups p : E→X whose fiber G is abelian, an action of π1(X) on G by automorphisms is the same as a Z[π1X] module structure on G. Proposition 3H.4. If X is a path-connected space having a universal covering space, th... |
1 + g2), the term g1 + g2 should be interpreted not in Z[G] σ ⊗ g1 + ∆ but in the natural quotient G of Z[G]. Hence Cn(X; E) is identified with the quoe e X) ⊗π Z[G]. This natural identification commutes with the X) ⊗π G of Cn( tient Cn( ⊔⊓ boundary homomorphisms, so the homology groups are also identified. σ ⊗ γ g. This ... |
1)→Hn(X; G2) for ordinary homology are a special case. To avoid this extra complication we shall e e consider only the case that f restricts to an isomorphism on each fiber. With this condition, a commutative diagram as above will be called a bundle map. Here is a method for constructing bundle maps. Starting with a map... |
that the map E→f ∗(E′), e ֏ (p(e), so the pullback construction produces all bundle maps. Thus we see one reason why homology with local coefficients is somewhat complicated: Hn(X; E) is really a functor of two variables, covariant in X and contravariant in E. e Viewing bundles of abelian groups over X as Z[π1X] modules... |
reader. ∆ Now we turn to cohomology. One might try defining H n(X; E) by simply dualizing, taking Hom(Cn(X), E), but this makes no sense since E is not a group. Instead, the cochain group C n(X; E) is defined to consist of all functions ϕ assigning In case E is the product to each singular simplex σ : n→X a lift ϕ(σ ) :... |
in §3.1 extend without great dif- e e e ∆ In order to define the map ficulty to cohomology groups with local coefficients. f ∗ : H n(X ′; E′)→H n(X; E) induced by a bundle map as before, it suffices to observe σ ′ : σ = that a singular simplex σ : n→f ∗(E) of σ. To show that f ≃ g implies f ∗ = g∗ requires some mod(σ, e ific... |
ology e For example, consider the n dimensional torus T n, the product of n circles, with e fundamental group π = Zn and universal cover Rn. We have Hi(T n; Z[π ]) ≈ Hi(Rn), which is zero except for a Z in dimension 0, but H i(T n; Z[π ]) ≈ H i c(Rn) vanishes except for a Z in dimension n, as we saw in Example 3.34. To... |
H n X n−1 in X n obtained by deleting an open n disk from the interior of each n cell. If X is locally compact, the obvious deformation retraction of N(X n−1) onto X n−1 is a proper homotopy equivalence. Hence via long exact sequences and the five-lemma c (X n, N(X n−1); G), and by excision the we obtain isomorphisms H... |
Cn, Z[π ]) are isomorphisms of f ( X; Z) and H n(X; Z[π ]) cochain complexes, so the respective cohomology groups H n c ( ⊔⊓ are isomorphic. b γ ϕ(γ−1ηen)γ by ηγ, we get ϕ defines a homomorphism C n f ( ϕ(en Cup and cap product work easily with local coefficients in a bundle of rings, the latter concept being defined in th... |
] so they split as the direct sum of their real and imaginary parts, which are just the homology or cohomology groups with ordinary coefficients Z and Z) constructed in Z, respectively. The fundamental class in Hn(M; twisted coefficients Example 3H.3 can be viewed as a pure imaginary fundamental class [M] ∈ Hn(M; E). Since... |
zero unless n = 1, when it is Z when g = 1 and the direct sum of a countably infinite number of Z ’s when g > 1. [Use Proposition 3H.5 and compute X) as lim--→H n( H n X − Ti) for a suitable sequence of finite subtrees T1 ⊂ T2 ⊂ ··· c ( i Ti = X with of e 6. Show that homology groups H ℓf n (X; G) can be defined using lo... |
usually false, however. One of the rare cases when a CW complex does have its homotopy type uniquely determined by its homo- topy groups is when it has just a single nontrivial homotopy group. Such spaces, known as Eilenberg–MacLane spaces, turn out to play a fundamental role in algebraic topology for a variety of rea... |
an arbitrary CW complex is built up from its homotopy groups by an inductive procedure of forming ‘twisted products’ of Eilenberg–MacLane spaces. This is the notion of a Postnikov tower. In favorable cases, including all simply-connected CW complexes, the additional data beyond homotopy groups needed to determine a ho... |
is the large region of zeros below the diagonal, and indeed πi(S n) = 0 for all i < n as we show in Corollary 4.9. There is also the sequence of zeros in the first row, suggesting that πi(S 1) = 0 for all i > 1. This too is a fairly elementary fact, a special case of Proposition 4.1, following easily from covering spac... |
otopy groups of spheres resides in finite abelian groups. The problem thus reduces to computing the p torsion in πi(S n) for each prime p. 340 Chapter 4 Homotopy Theory An especially interesting feature of the table is that along each diagonal the groups πn+k(S n) with k fixed and varying n eventually become independent ... |
= f (1 − s1, s2, ···, sn). The additive notation for the group operation is used because πn(X, x0) is abelian for n ≥ 2. Namely, f + g ≃ g + f via the homotopy indicated in the following figures. The homotopy begins by shrinking the domains of f and g to smaller subcubes of In, with the region outside these subcubes ma... |
epoint x1 = γ(1), we may associate to each map f : (In, ∂In)→(X, x1) a new map γf : (In, ∂In)→(X, x0) by shrinking the domain of f to a smaller concentric cube in In, then inserting the path γ on each radial segment in the shell between this smaller cube and ∂In. When n = 1 the map γf is the composition of the three pa... |
define a change-of-basepoint transformation βγ : πn(X, x1)→πn(X, x0) by βγ([f ]) = [γf ], then (1) shows that βγ is a homomorphism, while (2) and (3) imply that βγ is an isomorphism with inverse βγ where γ is the inverse path of γ, 342 Chapter 4 Homotopy Theory γ(s) = γ(1 − s). Thus if X is path-connected, different cho... |
elian if it has trivial action of π1 on all homotopy groups πn, since when n = 1 this is the condition that π1 be abelian. This terminology is consistent with a long-established usage of the term ‘nilpotent’ to refer to spaces with nilpotent π1 and nilpotent action of π1 on all higher homotopy groups; see [Hilton, Misl... |
n, s0)→(X, x0) lifts to ( x0) provided that n ≥ 2 so that S n is simply-connected. Injectivity of p∗ is immediate from the covering homotopy ⊔⊓ property, just as in Proposition 1.31 which treated the case n = 1. X, e e In particular, πn(X, x0) = 0 for n ≥ 2 whenever X has a contractible universal cover. This applies f... |
. There does not seem to be a completely satisfactory way of defining π0(X, A, x0), so we shall leave this undefined (but see the exercises for one possible definition). Note that πn(X, x0, x0) = πn(X, x0), so absolute homotopy groups are a special case of relative homotopy groups. A sum operation is defined in πn(X, A, x0... |
x0) iff it is homotopic rel S n−1 to a map with image contained in A. For if we have such a homotopy to a map g, then [f ] = [g] in πn(X, A, x0), and [g] = 0 via the homotopy obtained by composing g with a deformation retraction of Dn onto s0. Conversely, if [f ] = 0 via a homotopy F : Dn × I→X, then by restricting F t... |
by restricting maps (Dn, S n−1, s0)→(X, A, x0) to S n−1. The map ∂, called the boundary map, is a homomorphism when n > 1. Theorem 4.3. This sequence is exact. Near the end of the sequence, where group structures are not defined, exactness still makes sense: The image of one map is the kernel of the next, those element... |
a map (In, ∂In, J n−1)→(X, B, x0) to In−1 has image lying in B, and hence represents zero in πn−1(A, B, x0). Conversely, suppose the restriction of f : (In, ∂In, J n−1)→(X, A, x0) to In−1 represents zero in πn−1(A, B, x0). Then f || In−1 is homotopic to a map with image in B via a homotopy F : In−1 × I→A rel ∂In−1. We... |
etrizing the nth and (n + 1) st coordinates as shown in the second picture, we see that f with g tacked on is in the image of ∂. But as we noted in the preceding paragraph, tacking g onto f gives the same element of πn(A, B, x0). ⊔⊓ Example 4.4. Let CX be the cone on a path-connected space X, the quotient space of X × ... |
) is independent of x0 when A is path-connected. In this case πn(X, A, x0) is often written simply as πn(X, A). Restricting to loops at the basepoint, the association γ ֏ βγ defines an action of π1(A, x0) on πn(X, A, x0) analogous to the action of π1(X, x0) on πn(X, x0) in the absolute case. In fact, it is not hard to s... |
did not define the relative π0, and (1)–(3) are each equivalent to saying that each path-component of X contains points in A since D0 is a point and ∂D0 is empty. The pair (X, A) is called n connected if (1)–(4) hold for all i ≤ n, i > 0, and (1)–(3) hold for i = 0. Note that X is n connected iff (X, x0) is n connected ... |
rel ∂Dk in view of the hypothesis f that πk(Y, B, y0) = 0 if k > 0, or that (Y, B) is 0 connected if k = 0. This homotopy Φ induces a homotopy of f on the quotient space X k−1 ∪ ek of X k−1 ∐ Dk, a of f homotopy rel X k−1. Doing this for all k cells of X − A simultaneously, and taking the constant homotopy on A, we ob... |
raction Mf →Y. Since this retraction is a homotopy equivalence, it suffices to show that Mf deformation retracts onto X if f induces isomorphisms on homotopy groups, or equivalently, if the relative groups πn(Mf, X) are all zero. If the map f happens to be cellular, taking the n skeleton of X to the n skeleton of Y for a... |
× RP∞ having nonvanishing homology in infinitely many dimensions since it retracts onto RP∞. Another pair of CW complexes that are not homotopy equivalent but have isomorphic homotopy groups is S 2 and S 3 × CP∞, as we shall see in Example 4.51. In fact it turns out to be quite rare that the homotopy type of a CW compl... |
this and many other purposes in homotopy theory to require just that cells map to cells of the same or lower dimension. Such a map f : X→Y, satisfying f (X n) ⊂ Y n for all n, is called Homotopy Groups Section 4.1 349 a cellular map. It is a fundamental fact that arbitrary maps can always be deformed to be cellular. T... |
> n, otherwise f is already cellular on en. We will show below that it is possible to deform f || X n−1 ∪ en, staying fixed on X n−1, so that f (en) misses some point p ∈ ek. Then we can deform f || X n−1 ∪ en rel X n−1 so that f (en) misses the whole cell ek by composing with a deformation retraction of Y k − {p} onto... |
, where Z is obtained from a subspace W by rel f −1(W ) attaching a cell ek. Then there is a homotopy ft : from f = f0 to a map f1 for which there is a polyhedron K ⊂ In such that : (a) f1(K) ⊂ ek and f1 || K is PL with respect to some identification of ek with Rk. (b) K ⊃ f −1 1 (U) for some nonempty open set U in ek. ... |
chosen smaller than half the distance between the compact sets f −1(B1) and In − f −1(int(B2)), and then we will have K2 ⊂ f −1(B2). Homotopy Groups Section 4.1 351 Now we subdivide all the cubes of K2 into simplices. This can be done inductively. The boundary of each cube is a union of cubes of one lower dimension, s... |
2 by the choice of ε and the fact that K2 ⊂ f −1(B2). Since f (σ ) ⊂ Bσ and Bσ is convex, we must have g(σ ) ⊂ Bσ, hence also ft(σ ) ⊂ Bσ for all t, and in particular f1(σ ) ⊂ Bσ. We know that Bσ is not contained in B1 since σ contains points outside K1 hence outside f −1(B1). The radius of Bσ is half that ⊔⊓ of B1, so... |
that a weak homotopy equivalence Y, f (x0) between CW complexes is a homotopy equivalence. It follows easily that this holds also for spaces homotopy equivalent to CW complexes. In general, however, weak homotopy equivalence is strictly weaker than homotopy equivalence. For example, there exist noncontractible spaces ... |
Choose maps ϕα : (S k−1, s0)→(A, aγ) representing all nontrivial elements of the kernels of the maps f∗ : πk−1(A, aγ)→πk−1(X, f (aγ )) for all the basepoints aγ. We may assume the maps ϕα are cellular, where S k−1 has its standard CW structure with s0 as 0 cell. Attaching cells ek α to A via the maps ϕα then produces ... |
does it destroy surjectivity on πk−1 or indeed any πi, obviously. Now to construct a CW approximation f : Z→X one can start with A consisting of one point for each path-component of X, with f : A→X mapping each of these points to the corresponding path-component. Having now a bijection on π0, attach 1 cells to A to cr... |
n is also surjective since this is true for the composition f-----→ X by the hypothesis that (X, A) is n connected. In dimensions below n, f A֓Z induces isomorphisms on homotopy groups since both inclusions A֓ Z and A֓ X do. Thus f is a weak homotopy equivalence, and hence a homotopy equivalence. To see that f is a hom... |
and greater, and πi(Xn) = 0 for i > n so we can apply Lemma 4.7, the extension lemma. Thus we have a commu- tative diagram as at the right. This is a called a Postnikov tower for X. One can regard the spaces Xn as truncations of X which provide successively better approximations to X as n increases. Postnikov towers t... |
. Our earlier construction shows: Proposition 4.17. For every pair (X, A) with A a nonempty CW complex there exist n connected CW models f : (Z, A)→(X, A) for all n ≥ 0, and these models can be chosen to have the additional property that Z is obtained from A by attaching cells of dimension greater than n. ⊔⊓ The constr... |
to a map h with image in Z ′, by the compression lemma and the hypothesis n ≥ n′. This proves the first assertion. For the second, suppose h0 and h1 are two maps Z→Z ′ whose compositions with f ′ are homotopic to gf rel A. Thus if we regard h0 and h1 as maps to W, they are homotopic rel A. Such a homotopy gives a map (... |
with Zn n connected and the map Zn→X inducing an isomorphism on all homotopy groups πi with i > n. The space Z0 is path-connected and homotopy equivalent to the component of X containing A, so one may as well assume Z0 equals this component. The next space Z1 is simply-connected, and the map Z1→X has the homotopy prop... |
� Homotopy Groups Section 4.1 357 α in K, with ∂ L ⊂ K be the subcomplex consisting of (k − 1) simplices corresponding to the singular (k − 1) simplices in ∂α, so σ (L) ⊂ X. The chain α is the image under the chain map σ♯ of a chain α a chain in L. In relative homology we then α] = [α]. If we assume πi(Z, X) = 0 for i ... |
just spheres. The following proposition gives a precise statement, using the notations [X, Y ] for the set of homotopy classes of maps X→Y and hX, Y i for the set of basepointpreserving-homotopy classes of basepoint-preserving maps X→Y. (The notation hX, Y i is not standard, but is intended to suggest ‘pointed homotop... |
, and deduce that f +′ k ≃ f + k so the two sums agree on πn(X, x0), and also that g +′ h ≃ h + g so the addition is abelian. 2. Show that if ϕ : X→Y is a homotopy equivalence, then the induced homomorphisms ϕ∗ : πn(X, x0)→πn(Y, ϕ(x0)) are isomorphisms for all n. [The case n = 1 is Proposition 1.18.] 3. For an H–space ... |
for all n > 1. e e A = p−1(A), show that the e e e 7. Extend the results proved near the beginning of this section for the change-ofbasepoint maps βγ to the case of relative homotopy groups. ∂-----→ π0(A, x0) -→ π0(X, x0) is exact. 8. Show the sequence π1(X, x0) -→ π1(X, A, x0), 9. Suppose we define π0(X, A, x0) to be ... |
, the union of the iterated suspensions S nX. 12. Show that an n connected, n dimensional CW complex is contractible. 13. Use the extension lemma to show that a CW complex retracts onto any contractible subcomplex. 14. Use cellular approximation to show that the n skeletons of homotopy equivalent CW complexes without c... |
Y have a common CW approximation. 20. Show that [X, Y ] is finite if X is a finite connected CW complex and πi(Y ) is finite for i ≤ dim X. 21. For this problem it is convenient to use the notations X n for the nth stage in a Postnikov tower for X and Xm for an (m − 1) connected covering of X, where X is a connected CW c... |
. Excision for Homotopy Groups What makes homotopy groups so much harder to compute than homology groups is the failure of the excision property. However, there is a certain dimension range, depending on connectivities, in which excision does hold for homotopy groups: Theorem 4.23. Let X be a CW complex decomposed as t... |
sequence of suspension maps π1(S 1)→π2(S 2)→π3(S 3)→ ··· the first map is surjective and all the subsequent maps are isomorphisms. Since π1(S 1) is Z generated by the identity map, it follows that πn(S n) for n ≥ 2 is a finite or infinite cyclic group independent of n, generated by the identity map. The fact that this cy... |
which f is the restriction of a linear map from Ri to Rm+1 or Rn+1. We may assume these linear maps are surjections by rechoosing smaller simplices and n+1 in the complement of the images of the nonsurjective linear maps. ⊂ em+1 α m+1 α m+1 α m+1 α and ∆ ∆ ∆ ∆ α α ∆ Claim: If i ≤ m+n, then there exist points pα ∈ ∆ n+... |
−1(pα) for each α, but also so that f −1(q) and f −1(pα) have disjoint images under the projection π : Ii→Ii−1. This is equivalent to saying that f −1(pα) is disjoint from, the union of all segments {x}× I meeting f −1(q). This set T is T = π −1 a finite union of convex polyhedra of dimension ≤ i − n since f −1(q) is a... |
1(pα) as before. This allows us to excise F −1(q) from the domain of F, from which it follows that f0 and f1 represent the same element of πi(A, C, x0). Since Ii × I now plays the role of Ii, the dimension i is replaced by i + 1 and the dimension restriction i ≤ m + n becomes i + 1 ≤ m + n, or i < m + n. Case 2: A is o... |
vertical maps are isomorphisms by Case 2, while by induction the second and fifth maps are isomorphisms, so the middle map is an isomorphism by the five-lemma. Similarly, when i = m + n the second and fourth maps are surjective and the fifth map is injective, which is enough to imply the middle map is surjective by one h... |
induces an isomorphism on πn if n ≥ 2. By Proposition 4.2 we have α ), a free abelian group with basis the inclusions S n α πn(S n α S n α ) ≈ α, πn( Q W so the same is true for α ֓ α. This takes care of the case of finitely many S n α ’s. Q Q To reduce the case of infinitely many summands S n α S n W α πn(S n is surjec... |
(S 1 ∨ S n)] module on a single basis element, the homotopy class of the inclusion S n ֓ S 1 ∨ S n. Writing a generator of π1(S 1 ∨ S n) as t, the group ring Z[π1(S 1 ∨ S n)] becomes Z[t, t−1], the Laurent polynomials in t and t−1 with Z coefficients, and we have πn(S 1 ∨ S n) ≈ Z[t, t−1]. k S n W W This example shows th... |
utative diagram where the vertical isomorphism comes from a long exact sequence. Now apply the excision theorem to the first map in the diagram, using the fact that (CA, A) is (s + 1) connected if A is s connected, which comes from the exact sequence for the pair (CA, A). ⊔⊓ Example 4.29. Suppose X is obtained from a we... |
= 0 for i > 1 was replaced by the condition that X have a contractible universal cover, which is equivalent for spaces that have a universal cover of the homotopy type of a CW complex. We can build a CW complex K(G, n) for arbitrary G and n, assuming G is abelian if n > 1, in the following way. To begin, let X be an (... |
Having shown the existence of K(G, n) ’s, we now consider the uniqueness ques- tion, which has the nicest possible answer: Proposition 4.30. The homotopy type of a CW complex K(G, n) is uniquely determined by G and n. The proof will be based on a more technical statement: Lemma 4.31. Let X be a CW complex of the form ... |
mma by attaching cells of dimension n + 2 and greater. By the lemma there is a map f : X→K′ inducing an isomorphism on πn. To extend this f over K we proceed inductively. For each cell en+2, the composition of its attaching map with f is nullhomotopic in K′ since πn+1(K′) = 0, so f extends over this cell. The same argu... |
). e In the part of the theorem dealing with relative groups, notice that X must be e e e simply-connected as well as A since (X, A) is 1 connected by hypothesis. There is a more general version of the relative Hurewicz theorem given later in Theorem 4.37 that allows A and X to be nonsimply-connected, but this requires... |
have Hn(X) ≈ Coker d. ⊔⊓ β Z→ S W L e β β Since homology groups are usually more computable than homotopy groups, the following version of Whitehead’s theorem is often easier to apply: Corollary 4.33. A map f : X→Y between simply-connected CW complexes is a homotopy equivalence if f∗ : Hn(X)→Hn(Y ) is an isomorphism f... |
direct argument which avoids naturality questions. For the mapping cylinder Mf we know that πi(Mf, X) = 0 for i ≤ n. If this held also for i = n + 1 then the relative Hurewicz theorem would say that Hi(Mf, X) = 0 for i ≤ n + 1 and hence that f∗ would be an isomorphism on Hn. To make this argument work, let us temporar... |
πn(X) ≈ Z[t, t−1]/(2t − 1), where (2t − 1) denotes the ideal in Z[t, t−1] generated by 2t − 1. Note that setting t = 1/2 embeds Z[t, t−1]/(2t − 1) in Q as the subring Z[1/2] consisting of rationals with denominator a power of 2. From the long exact sequence of homotopy groups for the (n − 1) connected pair (X, S 1) we... |
α) where α is a fixed generator of Hn(Dn, ∂Dn) ≈ Z and f∗ : Hn(Dn, ∂Dn)→Hn(X, A) is induced by f. If we have a homotopy f ≃ g through maps (Dn, ∂Dn, s0)→(X, A, x0), or even through maps (Dn, ∂Dn)→(X, A) not preserving the basepoint, then f∗ = g∗, so h is well-defined. Proposition 4.36. The Hurewicz map h : πn(X, A, x0)→H... |
diagonal map x ֏ (x, x). From the equalities (f ∨ g)i1 = f and (f ∨ g)i2 = g we deduce that (f ∨ g)∗(i1∗ + i2∗) sends (x, 0) to f∗(x) and (0, x) to g∗(x), hence (x, x) to f∗(x) + g∗(x). Thus the composition across the top of the diagram is 370 Chapter 4 Homotopy Theory x ֏ f∗(x) + g∗(x). On the other hand, f + g = (f ... |
) if π1(X, x0) acts nontrivially on πn(X, x0), namely elements of the form [γ][f ]−[f ]. This is because γf and f, viewed as maps S n→X, are homotopic if we do not require the basepoint to be fixed during the homotopy, so (γf )∗(α) = f∗(α) for α a generator of Hn(S n). Similarly in the relative case the kernel of h : πn... |
� Hi(X, A) = 0 for i < n. Note that this statement includes the absolute form of the theorem by taking A to be the basepoint. Before starting the proof of this general Hurewicz theorem we have a preliminary step: α are attached for α, then πn(W, X) is a free α of the α, provided that the map π1(X)→π1(W ) induced by inc... |
. The inclusion (W, X)֓(W, Z) is a homotopy equivalence of pairs. Homotopy excision W gives a surjection π2(Y, Y 1)→π2(W, Z). The universal cover Y of Y is obtained from αβ of the disks D2 α. the universal cover α. Let Y = X α int(D2 X of X by taking the wedge sum with lifts e α, and let Z = W − α D2 D2 S Hence we have... |
first and third rows are exact sequences for the triple (X, X n ∪ A, A). To construct the map ∂′ would take a small extra argument but we will not actually need this map so it can be ignored for the proof. If we did have this map and we knew the middle row was exact, the theorem would follow from the five-lemma once we ... |
(y) for some y ∈ π ′ n(X n∪A, A). We have i∗h′(y) = 0 in the lower row so h′(y) comes from an element of Hn+1(X, X n ∪ A) hence also from some z ∈ πn+1(X, X n ∪ A) since the maps in the left column are surjective. We have h′q∂(z) = h′(y) by commutativity, hence q∂(z) = y since the middle h′ is injective. Then x = i′ ∗q... |
ology groups. Proof: Choose loops ϕα : S 1→X 1 generating π1(X) and use these to attach cells e2 to X to form a simply-connected CW complex X ′. The homology exact sequence α 0 -→ H2(X) -→ H2(X ′) -→ H2(X ′, X) -→ 0 = H1(X) splits since H2(X ′, X) is free with basis the cells e2 α. Thus we have an isomorphism H2(X ′) ≈... |
) = H∗( on homology. X we X) = 0, so the map X→X + induces an isomorphism X +/ X +, e e This construction X→X +, killing a perfect subgroup of π1(X) while preserving homology, is known as the Quillen plus construction. In some of the main applications X is a K(G, 1) where G has perfect commutator subgroup, so the map X... |
−1(b) ⊂ E, which are called fibers, are homeomorphic. For example, E could be the product F × B with p : E→B the projection. General fiber bundles can be thought of as twisted products. Familiar examples are the M¨obius band, which is a twisted annulus with line segments as fibers, and the Klein bottle, which is a twisted... |
fining the property that leads to a long exact sequence of homotopy groups. A map p : E→B is said to have the homotopy lifting property with respect to a space X if, given a homotopy gt : X→B gt : X→E and a map lifting gt. From a formal point of view, this can be regarded as a special case of the lift extension property... |
. In other words, the homotopy lifting property for extending a given lift (X, A) is the lift extension property for (X × I, X × {0} ∪ A× I). gt : X→E starting with a given lift e e The homotopy lifting property for Dk is equivalent to the homotopy lifting property for (Dk, ∂Dk) since the pairs (Dk × I, Dk × {0}) and (... |
� is similar. Given f0]) = p∗([ f1. p∗([ f1 on In × {1}, and the constant map We have a partial lift e to x0 on J n−1 × I. After permuting the last two coordinates of In × I, the relative hoG : In × I→E. motopy lifting property gives an extension of this partial lift to a full lift f1. So p∗ is injective. e For the las... |
as above is called a local trivialization of the bundle. Since the first coordinate of h is just p, h is determined by its second coordinate, a map p−1(U)→F which is a homeomorphism on each fiber Fb. The fiber bundle structure is determined by the projection map p : E→B, but to indicate what the fiber is we sometimes writ... |
, ···, zn) to its equivalence class [z0, ···, zn], so the fibers are copies of S 1. To see that the local triviality condition for fiber bundles is satisfied, let Ui ⊂ CPn be the open set of equivalence classes [z0, ···, zn] with zi ≠ 0. Define hi : p−1(Ui)→Ui × S 1 by hi(z0, ···, zn) = ([z0, ···, zn], zi/|zi|). This takes... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.