text stringlengths 270 6.81k |
|---|
�s for 1 ≤ i ≤ n form an additive basis for H ∗(Fn(C∞); Z), hence this ring is the polynomial ring on the xi ’s. There is a corresponding result for Fn(Ck), that H ∗(Fn(Ck); Z) is free with basis the monomials xi1 n with ij ≤ k−j for each j. This is proved in exactly the same way, using induction on n and the fiber bund... |
p∗ = p∗, which says that polynomials in the image of p∗ are invariant under permutations of the variables. As remarked earlier, the symmetric polynomials in Z[x1, ···, xn] form a polynomial ring Z[σ1, ···, σn] where σi has degree i. We have shown that the image of p∗ is a direct summand, so to show that p∗ maps onto th... |
Z) we see that the Poincar´e series p(t) of H ∗(Gn(C∞); Z) satisfies p(t)(1 − t)−n (1 − ti) = (1 − t)−n and hence p(t) = n Yi=1 (1 − ti)−1 n Yi=1 This is exactly the Poincar´e series of Z[σ1, ···, σn] since σi has degree i. As noted before, this implies that the image of p∗ is all the symmetric polynomials. This finishe... |
R) -→ H i−n+1(B; R) -→ ··· where e is a certain ‘Euler class’ in H n(B; R). Since H i(B; R) = 0 for i < 0, the ≈-----→ H i(E; R) initial portion of the Gysin sequence gives isomorphisms p∗ : H i(B; R) for i < n − 1, and the more interesting part of the sequence begins p∗------------→ H n−1(E; R) -→ H 0(B; R) p∗-------... |
1) has the same cohomology as the product S n−1 × S n if n is odd, at least when n > 1 so that the base space S n is simply-connected and the orientability hypothesis is satisfied. When n is even, the calculations at the end of §3.D show that H ∗(V2(Rn+1); Z) consists of Z ’s in dimensions 0 and 2n − 1 and a Z2 in dimen... |
1 we take Z2 coefficients. If n > 1 then since E is contractible, the Gysin sequence implies that H i(B; Z) = 0 for 0 < i < n and that `e : H i(B; Z)→H i+n(B; Z) is an isomorphism for i ≥ 0. It follows that H ∗(B; Z) is the polynomial ring Z[e]. When n = 1 the map p∗ : H n−1(B; Z2)→H n−1(E; Z2) in the Gysin sequence is ... |
this section, so e = w1 and the map `e : H ∗(Gn; Z2)→H ∗(Gn; Z2) is injective. The Gysin sequence then breaks `e------------→ H i+1(Gn; Z2)→H i+1( up into short exact sequences 0→H i(Gn; Z2) Gn; Z2)→0, from which it follows that H ∗( Gn; Z2) is the quotient ring Z2[w1, ···, wn]/(w1) ≈ Z2[w2, ···, wn]. `e------------→ ... |
2 connected, so e must be a generator of and this last group is zero since H 2(Gn; Z) ≈ Z. Since H ∗(Gn; Z) is a polynomial algebra Z[c1, ···, cn], we must have e = ±c1, so the map `e : H ∗(Gn; Z)→H ∗(Gn; Z) is injective, the Gysin sequence breaks up into short exact sequences, and H ∗( Gn; Z) is the quotient ring Z[c... |
� F, with local trivializations for E′ obtained by restricting local trivializations for E. For example, if E→B is a bundle with fiber Dn and E′ ⊂ E is the union of the boundary spheres of the fibers, then (E, E′) is a fiber bundle pair since local trivializations of E restrict to local trivializations of E′, in view of t... |
; R) ≈ H ∗(E, E′; R) via excision and the obvious have H ∗( E − B onto E. The long exact sequence of a triple gives deformation retraction of b H ∗( b are H ∗(B; R) module isomorphisms. Since B is a retract of E→B, we have a splitting H ∗( Let b The classes b F ∪ CF ′, so the absolute form of the Leray–Hirsch theorem i... |
; Z)→H n(E, E′; R) induced by the homomorphism Z→R sending 1 to the identity element of R. Corollary 4D.9. If the disk bundle (Dn, S n−1) -→ (E, E′) : H i(B; R)→H i+n(E, E′; R), c ∈ H n(E, E′; R), then the map an isomorphism for all i ≥ 0, and H i(E, E′; R) = 0 for i < n. p-----→ B has a Thom class (b) = p∗(b) ` c, is ... |
fiber is a CW complex since the homotopy lifting property was used only for the fiber and for the product of the fiber with I. In the case of a sphere bundle S n−1→E′→B, if γ is a loop in B then Lγ is a homotopy equivalence from the fiber S n−1 over the basepoint of γ to itself, and we define the sphere bundle to be orient... |
is also true: A disk bundle is orientable if it has a Thom class with Z coefficients. Proof: The case of a non-CW base space B reduces to the CW case by pulling back over a CW approximation to B, as in the Leray–Hirsch theorem, applying the five-lemma to say that the pullback bundle has isomorphic homotopy groups, hence ... |
−1 is homotopic to the composition of Lγ with the inclusion (Dn will use this preferred isomorphism H n(E, E′; Z) ≈ Z in the inductive proof of (∗) In the case of Z2 coefficients, there can be only one isomorphism of given below. a group with Z2 so no choices are necessary and orientability is irrelevant. We will prove (... |
�)−1j∗(c), c being a Thom class. The square containing the map `e commutes since for b ∈ H i−n(B; R) we have j∗ (b) = j∗(p∗(b) ` c) = p∗(b) ` j∗(c), which equals p∗(b ` e) = p∗(b) ` p∗(e) since p∗(e) = j∗(c). Another way of defining e Φ is as the class corresponding to c ` c under the Thom isomorphism, since (e) = p∗(e)... |
4.D 445 Since p∗ is surjective, we can choose elements yj ∈ H ∗(B; R) with p∗(yj ) = xj. It remains to check that H ∗(B; R) = R[y1, ···, yℓ, e], which is elementary algebra: Given b ∈ H ∗(B; R), p∗(b) must be a polynomial f (x1, ···, xℓ), so b − f (y1, ···, yℓ) is in the kernel of p∗ and exactness gives an equation b ... |
0 in H ∗(B; R) then has the form g(y1, ···, yℓ, e) ` e = 0. Since `e is injective, this gives a polynomial relation g(y1, ···, yℓ, e) = 0 with g having lower degree than f. By induction we deduce that g must be the zero polynomial, hence also f. ⊔⊓ Example 4D.12. Let us apply this to give another proof that H ∗(Gn(C∞)... |
of [VBKT]. Before giving our next example, let us observe that the Gysin sequence with a p¬-----→ B whose fixed coefficient ring R is valid for any orientable fiber bundle F -→ E fiber is a CW complex F with H ∗(F ; R) ≈ H ∗(S n−1; R). Orientability is defined just γ : H n−1(F ; R)→H n−1(F ; R). No changes are as before in ... |
a positively oriented basis for P. Both bundles Gn(R∞) are simply-connected, from the bundle e are orientable since their base spaces SO(n)→Vn(R∞)→ Gn(R∞). e e The fiber V2(R∞) of the second bundle is contractible, so E has the same cohoG2k−1(R∞). The fiber of the first bundle has the same Zp cohomology as mology as S 4k... |
−1, e], e |pi| = 4i |pi| = 4i, |e| = 2k The same result holds also with Q coefficients. e In fact, our proof applies for any coefficient ring in which 2 has a multiplicative inverse, since all that is needed is that Gn(R∞) H ∗(V2(R2k+1); R) ≈ H ∗(S 4k−1; R). For a calculation of the cohomology of with Z coefficients, see [VB... |
≈ Z[x1, ···, xn]/(σ1, ···, σn) Λ where σi is the ith elementary symmetric polynomial. 5. Use the Gysin sequence to show that for a fiber bundle S k→Sm p-----→ S n we must have k = n − 1 and m = 2n − 1. Then use the Thom isomorphism to show that the Hopf invariant of p must be ±1. [Hence n = 1, 2, 4, 8 by Adams’ theorem... |
is connected.] H n+i( Σ e e 10. Fill in the details of the following argument to show that every n× n matrix (The usual argument over C involvA with entries in H has an eigenvalue in H. ing roots of the characteristic polynomial does not work due to the lack of a good quaternionic determinant function.) For t ∈ [0, 1]... |
functors hn, n ∈ Z, from C to abelian groups, together with natural isomorphisms hn(X) ≈ hn+1( X) for all X in C, such that the following axioms hold for each hn : (i) If f ≃ g : X→Y in the basepointed sense, then f ∗ = g∗ : hn(Y )→hn(X). (ii) For each inclusion A֓X in C the sequence hn(X/A)→hn(X)→hn(A) is exact. α Xα... |
E 449 So let us consider what properties the functor h(X) = hX, Ki has, where K is a fixed space with basepoint. First of all, it is a contravariant functor from the category of basepointed CW complexes to the category of pointed sets, that is, sets with a distinguished element, the homotopy class of the constant map in... |
h, then, using the notation f : (K, u)→(K′, u′) to mean f : K→K′ with f ∗(u′) = u, universality implies that there are maps f : (K, u)→(K′, u′) and g : (K′, u′)→(K, u) that are unique up to homotopy. Likewise the compositions gf : (K, u)→(K, u) and f g : (K′, u′)→(K′, u′) are unique up to homotopy, hence are homotopic... |
), so the Mayer–Vietoris axiom implies there is an element of h(X ∪ CA) restricting to z in h(Z) and hence to x in h(X). (3) If h satisfies axioms (i) and (iii) then h( Y ) is a group and Tu : h Y, Ki→h( Y ) is a homomorphism for all suspensions Y and all pairs (K, u). [See the Corrections.] Σ Σ Σ The proof of Theorem ... |
this element restricts to the trivial element of h( α ) by the definition of f. The exactness property of h then implies that for the reduced mapping cone Cf = Mf / α there is an element w ∈ h(Cf ) restricting to un on Kn. Note that Cf is obtained from Kn by attaching cells en+1 by the maps fα. To finish the constructio... |
complexes Ki × [i, i+1] for i odd and let B be the corresponding union for i even. Thus Ki, A ≃ i K2i. By the wedge axiom there exist a ∈ h(A) and b ∈ h(B) restricting to ui on each Ki. Then using the fact that ui+1 || Ki = ui, the Mayer–Vietoris axiom implies that a and b are the restrictions of an element t ∈ h(T ). ... |
֓ K′ to a map g : X→K. The relation g∗(u) = x holds since u′ || K = u and u′ || X = x. ⊔⊓ Proof of Theorem 4E.2: It suffices to show that a π∗ universal pair (K, u) is universal. Applying the preceding lemma with A a point shows that Tu : hX, Ki→h(X) is surjective. To show injectivity, suppose Tu(f0) = Tu(f1), that is, ... |
nf, which says that the map Φ Kn+1i is composition with εn. Since is a bijection, if we take X to be S i, we see that εn induces an isomorphism on πi X) correspond to weak homotopy equivalences It remains to show that the natu- Σ X, Kn+1i = hX, X) corresponds to a nat- Kn+1i which we call. The naturality ( for all i, s... |
the preceding section we reformulated the axioms for a cohomology theory so that the exactness axiom asserts just the exactness of hn(X/A)→hn(X)→hn(A) for CW pairs (X, A). In order to derive long exact sequences, the reformulated axioms Spectra and Homology Theories Section 4.F 453 Σ require also that natural suspensi... |
((X) L A modest generalization of this homology theory can be obtained by defining hn(X) = π s n(X ∧ K) for a fixed complex K. Verifying the homology axioms reduces to the case of stable homotopy groups themselves by basic properties of smash product: X) since (X ∧ K) = ( hn(X) ≈ hn+1( The exactness axiom holds since (X ... |
G, n)→K(G, n + 1), or to write this more concisely, Kn→Kn+1 This induces a map π s i+n+1(X ∧ Kn+1). Via these maps, it then i+n+1(X ∧ Σ K(G, n+1). Since suspension plays such i+n(X ∧ Kn) = π s Kn)→π s Ω Σ Σ 454 Chapter 4 Homotopy Theory makes sense to consider the direct limit as n goes to infinity, the group hi(X) = li... |
same as the stable homotopy groups of X. For a general spectrum K we could also i+n(Kn) since the composition πi+n(Kn)→πi+n+j(Kn+j) facdescribe πi(K) as lim--→π s jKn). So the homotopy groups of a spectrum are ‘stable homotors through πi+n+j( topy groups’ essentially by definition. Σ Returning now to the context of hom... |
� Σ Spectra and Homology Theories Section 4.F 455 limit of a product need not equal the product of the direct limits. For finite wedge sums there is no difficulty, so we do have a cohomology theory for finite CW complexes. But for general CW complexes a different definition is needed. The simplest thing to do ni. We iKn+i, t... |
(QX) = π s S into ordinary homotopy groups. Ω Σ Ω Σ It follows routinely from the definitions that the homology theory defined by a spectrum is the same as the homology theory defined by the associated spectrum. One may ask whether every homology theory is defined by a spectrum, as we showed for cohomology. The answer is y... |
be the original topology on X. An action of a group G on a space X determines a diagram of spaces XG, with X itself as the only space and with maps the homeomorphisms g : X→X, g ∈ G, given by the action. In this case XG is the orbit space X/G. Gluing Constructions Section 4.G 457 complex X can be viewed as a diagram o... |
one map from this space to itself, then X is the mapping torus. For a diagram consisting of two maps f, g : X0→X1 the space X was studied in Ex∆ ample 2.48. Mapping telescopes are the case of a sequence of maps X0→X1→ ···. In §1.B we considered general diagrams in which the spaces are K(G, 1) ’s. g-----→ X2 the realiz... |
are the finite intersections of Xi ’s and whose edges are inclusions is a complex of for this complex of spaces with n simplices the n fold inclusions. The base spaces is the barycentric subdivision of the nerve of the cover. Recall from the end of §3.3 that the nerve of a cover is the simplicial complex with n simplic... |
it suffices to show that the inclusion M n−1( homotopy equivalence and the pair (M( X. It is enough to show this In this case f is a map from X0→ ··· →Xn to Y0→ ··· →Yn. By ∆ f ) is a X ֓ M( X) satisfies the homotopy ∆ extension property. The latter assertion is evident from Example 0.15 since a mapping f ), M n−1 ) lyin... |
section s embeds X as a retract of is a deformation retract since points in fibers p−1(x) can move ⊔⊓ linearly along line segments to s(x). ∆ Corollary 4G.3. If U is an open cover of a paracompact space X such that every nonempty intersection of finitely many sets in U is contractible, then X is homotopy equivalent to t... |
→ is a fiber bundle. 3. What is the nerve of the cover of a simplicial complex by the open stars of its Γ vertices? [See Lemma 2C.2.] ∆ 4. Show that Proposition 4G.2 and its corollary hold also for CW complexes and covers by families of subcomplexes. [CW complexes are paracompact; see [VBKT].] There is a very nice duali... |
omorphism onto its image A× {1 − t}, e gt : i(A)→A is a continuous inverse of the relation i : A→i(A). ⊔⊓ gti = gt implies that the map g−1 gt : B→Mi with e e t e e Many constructions for fibrations have analogs for cofibrations, and vice versa. For example, for an arbitrary map f : A→B the inclusion A ֓ Mf is readily se... |
strict topological analog of a kernel is a fiber of a fibration. Dually, the topological analog of a cokernel is the cofiber B/A of a cofibration A֓ B. If we make an arbitrary map f : A→B into a cofibration A ֓ Mf, the cofiber is the mapping cone Cf = Mf /(A× {0}). In the diagram showing Ef and Mf as pullback and pushout, t... |
and pullback constructions can be generalized to arbitrary diagrams. In Σ the case of pushouts, this was done in §4.G where we associated a space X to a diav Xv, with v ranging over gram of spaces X. This was the quotient of the coproduct vertices of the diagram, under the identifications x ∼ fe(x) for all maps fe asso... |
general diagrams as well. Homotopy Groups with Coefficients ∆ There is a somewhat deeper duality between homotopy groups and cohomology, which one can see in the fact that cohomology groups are homotopy classes of maps into a space with a single nonzero homotopy group, while homotopy groups are ho- motopy classes of map... |
for these groups µn(X; G) : Proposition 4H.2. For n > 1 there are natural short exact sequences 0 -→ Ext(G, πn+1(X)) -→ µn(X; G) -→ Hom(G, πn(X)) -→ 0. The similarity with the universal coefficient theorem for cohomology is apparent, but with a reversal of the variables in the Ext and Hom terms, reflecting the fact that ... |
M(Z2, n) 464 Chapter 4 Homotopy Theory defines an element of µn(M(Z2, n); Z2) = hM(Z2, n), M(Z2, n)i having order 4, as we show in Example 4L.7, whereas the two outer terms in the short exact sequence can only contain elements of order 2 since G = Z2. This example shows also that µn(X; Zm) need not be a module over Zm,... |
in fact special cases of a Hurewicz-type theorem relating πn(X; Zm) and Hn(X; Zm), which is proved in [Neisendorfer 1980]. Along with Z and Zm, another extremely useful coefficient group for homology and cohomology is Q. We pointed out above the difficulty that there is no cohomology analog of M(Q, n). The groups µn(X; Q)... |
hn : M(Gn+1, n)→Xn such that the induced map hn∗ : Hn (3) X = n Xn. M(Gn+1, n) →Hn(Xn) is trivial. S One sees inductively that Xn+1 has the desired homology groups by comparing the long exact sequences of the pairs (Xn+1, Xn) and (CM, M) where M = M(Gn+1, n) and CM is the cone M × I/M × {0} : The assumption that hn∗ i... |
� Hn+1(Y ) = Gn+1. We will use this isomorphism to construct a map hn : M(Gn+1, n)→Xn and an extension f : Xn+1→Y. The standard construction of an M(Gn+1, n) consists of a wedge of spheres α corresponding to generators gα of Gn+1, with cells en+1 S n attached according to certain linear combinations rβ = α nαβgα that a... |
CW complex having all its homology groups free. Then the Moore spaces used in the construction of X can be taken to be wedges of spheres, and so Xn is obtained from Xn−1 by attaching an n cell for each Z summand of Hn(Y ). The attaching maps may be taken to be cellular, making X into a CW complex whose cellular chain ... |
homotopy type, as the suspension becomes homotopy equivalent to a wedge sum of Σ Stable Splittings of Spaces Section 4.I 467 smaller spaces. Much of the interest in such stable splittings comes from the fact that they provide a geometric explanation for algebraic splittings of homology and cohomology groups, as well a... |
contractible, collapsing the cones gives a homotopy equivalence (X × Y ). Inside X ∗ Y there are also cones x0 ∗ Y and X ∗ y0 (X ∧ Y ) and X ∗ Y ∪ CX ∪ CY ≃ intersecting in a point. Collapsing these cones converts X ∗ Y into Σ Σ X ∗ Y ∪ CX ∪ CY into ( ⊔⊓ This result can be applied inductively to obtain splittings for ... |
(X) is X ∧k. Thus we have maps Σ X × k -→ Jk(X) -→ X ∧k = Jk(X)/Jk−1(X) By repeated application of the preceding proposition, X ∧k is a wedge summand of X × k, up to homotopy equivalence. The proof shows moreover that there is a map X ∧k→ X ∧k is homotopic to the Σ identity. This composition factors as Σ X × k such tha... |
·∨Xp−1 where Xi is a CW complex having nonzero only in dimensions congruent to 2i mod 2p − 2. Σ e This result is best possible in a strong sense: No matter how many times any one of the spaces Xi is suspended, it never becomes homotopy equivalent to a nontrivial wedge sum. This will be shown in Example 4L.3 by studying... |
Some of these summands occur more than once, as we see in the case of Z2 × Z2. Σ K(Z2, 1) ∧ K(Z2, 1) Σ Σ Exercises 1. If a connected CW complex X retracts onto a subcomplex A, show that X ≃ (X/A). [One approach: Show the map A ∨ (X/A) induces an isomorphism on homology, where r : X→A is the retraction and q : X→X/A is... |
be attributed to its close connection with the sus- ΩΣ pension homomorphism πi(X)→πi+1( lence of ΩΣ X with J(X) to give another proof that the suspension homomorphism is X). We will use the weak homotopy equiva- ΩΣ an isomorphism in dimensions up to approximately double the connectivity of X. In addition, we will obta... |
1 ··· xk) = λ(x1) ··· λ(xk), the product of the loops λ(xi). The only difficulty is in the parametrization of this product, which needs to be adjusted so that λ is continuous. The problem is that when some xi approaches the basepoint e ∈ S n, one wants the loop λ(xi) to disappear gradually from the product λ(x1) ··· λ(xk... |
X as the union of two reduced cones Σ Σ Ω Y+ = C+X and Y− = C−X intersecting in the equatorial X ⊂ X. Consider the path fibration p : P Y →Y with fiber Y. Let P+Y = p−1(Y+) and P−Y = p−1(Y−), so P+Y consists of paths in Y starting at the basepoint and ending in Y+, and similarly for P−Y. Then P+Y ∩ P−Y is p−1(X), the pa... |
ber homoY × Y+, and this is easily constructed as follows. Define topy equivalence P+Y ≃ f+ : P+Y → y is the obvious path in Y+ from y = (x, t) to the basepoint along the segment {x}× I. In the other y where the bar denotes the direction, define g+ : inverse path. Then f+g+ and g+f+ are fiber-homotopic to the respective i... |
an isomorphism 2 is an isomorphism. Via the gives an isomorphism L e Ω 1(γ, x) = Y ; F ). Since H∗(X; F ) H∗( Θ H∗( Θ 2. Hence Ω H∗( e from the first summand H∗( (γ λ(x)), this means that the composed map H∗(X; F ) onto Θ Θ H∗( Y ; F ) ⊗ H∗(X; F ) e Ω 11 ⊗ λ∗ -----------------------------→ H∗( e Ω Θ Y ; F ) ⊗ H∗( Y ; F... |
⊗ vj Proof: Since µ is an isomorphism, each element a ∈ An with n > 0 can be written j aji(vj ) for vj ∈ V and aj ∈ An(j), with uniquely in the form µ n(j) < n since V0 = 0. By induction on n, aj = i(αj) for a unique αj ∈ (T V )n(j). P Thus a = i so i is surjective. Since these representations are unique, i is also in... |
⊔⊓ ΩΣ Using the natural identification πi( induces the suspension map πi(X)→πi+1( ΩΣ J(X), we can identify the relative groups πi( n connected then the pair (J(X), X) is (2n + 1) connected since we can replace X by a complex with n skeleton a point, and then the (2n + 1) skeleton of J(X) is X). Since this inclusion fac... |
there is no map of Hopf invariant ±1, it follows that [11, 11] generates a Z summand of π2n−1(S n), and so the suspension homomorphism simply cancels this summand from π2n−1(S n). By Adams’ theorem, this is the situation for all even n except 2, 4, and 8. When n = 2 we have π3(S 2) ≈ Z generated by the Hopf map η with... |
n) in the same range i ≤ 3n − 2 since J(S n)/S n has S 2n as its (3n − 1) skeleton. Thus the terminal portion of the long exact sequence of the pair (J(S n), S n) starting with the term π3n−2(S n) can be written in the form π3n−2(S n) Σ-----→ π3n−1(S n+1)→π3n−2(S 2n)→π3n−3(S n) Σ-----→ π3n−2(S n+1)→ ··· This is known a... |
stable range. We show in Proposition 4L.11 2 is nonzero, and so we conclude that πn+2(S n) ≈ Z2 for all that the stable group π s n ≥ 2, generated by the composition ( n−2η)( We will see in [SSAT] that the EHP sequence extends all the way to the left to form n−1η). Σ an infinite exact sequence when n is odd, and when n... |
, n) to Eilenberg–MacLane spaces K(G, n). This leads to the H∗(X; Z) for all connected CW complexes X. e general result that for all connected CW complexes X, SP (X) has the homotopy type of a product of Eilenberg–MacLane spaces. In other words, the k invariants of SP (X) are all trivial. 476 Chapter 4 Homotopy Theory ... |
spaces A and B such that X is the union of the interiors of A and B. In this situation we call (X; A, B) an excisive triad. By a map f : (X; A, B)→(Y ; C, D) we mean f : X→Y with f (A) ⊂ C and f (B) ⊂ D. Proposition 4K.1. Let f : (X; A, B)→(Y ; C, D) be a map of excisive triads. If the induced maps πi(A, A ∩ B)→πi(C, C... |
→πi(Y, C) induced by inclusion is surjective for i = n and has trivial kernel for i = n − 1. (ii) Let ∂Dn be written as the union of hemispheres ∂+Dn and ∂−Dn intersecting in S n−2. Then every map (Dn × {0} ∪ ∂+Dn × I, ∂−Dn × {0} ∪ S n−2 × I) -→ (Y, C) taking (∂+Dn × {1}, S n−2 × {1}) to (X, A) extends to a map (Dn × I... |
(Y, C) by a map f : Dn × {0}→Y taking ∂−Dn × {0} to C and ∂+Dn × {0} to a chosen basepoint. Extend f over ∂+Dn × I via the constant homotopy, then extend over Dn × I by applying (iii). The result is a homotopy of the given f to a map representing an element of the image of πn(X, A)→πn(Y, C). Now we show that (i) implie... |
by its union with f −1(C)× (1/2, 1) in Mf, and enlarge Mf |B similarly. Now we prove the proposition for an inclusion (X; A, B) ֓ (Y ; C, D). The case n = 0 is trivial from the definitions, so let us assume n ≥ 1. In view of the equivalence of condition (i) with (ii) and (iii), it suffices to show that condition (ii) for... |
with image in X, independent of the I coordinate, and we may assume the condition (∗) holds here since we may assume that A = X ∩ C and B = X ∩ D, these conditions holding for the mapping cylinder construction described above. Consider the problem of extending f over K × I for K one of the subcubes. We may assume that... |
, U1)→πn(Y, V1) are isomorphisms. Hence by the five-lemma again, so are the maps πn(X)→πn(Y ). By induction, the case of finite covers by k > 2 sets reduces to the case of covers by two sets, by letting one of the two sets be the union of the first k − 1 of the given sets and the other be the k th set. The case of infinite... |
a weak homotopy equivalence. Recall that Fb is the space of all pairs (x, γ) with x ∈ E and γ a path in B from p(x) to b. The actual fiber p−1(b) is included in Fb as the pairs with x ∈ p−1(b) and γ the constant path at x. To see the equivalence of the two definitions, consider the commutative triangle at the right, whe... |
in some Bn, and such that each restriction p−1(Bn)→Bn is a quasifibration. (c) There is a deformation Ft of E into a subspace E′, covering a deformation F t of B into a subspace B′, such that the restriction E′→B′ is a quasifibration and F1 : p−1(b)→p−1 is a weak homotopy equivalence for each b ∈ B. By a ‘deformation’ i... |
i (b) Since each compact set in B lies in some Bn, each compact set in E lies in some subspace En = p−1(Bn), so πi En, p−1(b) just →πi(B, b) is an as πi(B, b) = lim--→πi(Bn, b). →πi(Bn, b) is an isomorphism isomorphism since each of the maps πi by assumption. We can take the point b to be an arbitrary point in B and th... |
� : SPn(X)→SPn(Y ), and f ≃ g implies f∗ ≃ g∗. Hence X ≃ Y implies SPn(X) ≃ SPn(Y ). In similar fashion SP is a homotopy functor on the category of basepointed spaces and basepoint-preserving homotopy classes of maps. It follows that X ≃ Y implies SP (X) ≃ SP (Y ) for connected CW complexes X and Y since in this case r... |
value f (a1, ···, an) is unchanged under permutation of the ai ’s, so there is an induced map SPn(S 2)→CPn which is a continuous bijection, hence a homeomorphism since both spaces are compact Hausdorff. Letting n go to ∞, we then get a homeomorphism SP (S 2) ≈ CP∞. The same argument can be used to show that SPn(S 1) ≃ ... |
ifications in all these copies of the boundary. The identifications on the boundary of Dn × Dn itself yield S n. This is clear since the identification (x, y) ∼ (y, x) converts Dn × ∂Dn ∪∂Dn × Dn to Dn × ∂Dn, and all points of ∂Dn are identified in S n. It remains to see that the identifications (x, y) ∼ (y, x) on each conc... |
This quotient is precisely S nRPn−1. This example illustrates that passing from a CW structure on X to a CW structure on SPn(X) or SP (X) is not at all straightforward. However, if X is a simplicial comcomplex structures on SPn(X) and SP (X), plex, there is a natural way of putting as follows. A simplicial complex str... |
result of this section, the Dold–Thom theorem: Theorem 4K.6. The functor X ֏ πiSP (X) for i ≥ 1 coincides with the functor X ֏ Hi(X; Z) on the category of basepointed connected CW complexes. In particular this says that SP (S n) is a K(Z, n), and more generally that for a Moore space M(G, n), SP (M(G, n)) is a K(G, n)... |
is the Hurewicz homomorphism, as we will see at the end of the proof of the Dold–Thom the- orem, implies that the map f induces an isomorphism on all homotopy groups. Thus →X, and as we noted above, n SP has only one we have a weak homotopy equivalence is a K(Gn, n). Finally, since each factor SP M(Gn, n) SP Q nontriv... |
we will show is that the projection SP (X)→SP (X/A) has instead the weaker structure of a quasifibration, which is still good enough to deduce a long exact se- quence of homotopy groups. Proof of 4K.6: As we have said, the main step will be to associate a long exact sequence of homotopy groups to each simplicial pair (... |
simply to slide points along the segments {a}× I in the mapping cylinder, with N = A× [0, 1/2). Let U ⊂ En consist of those points having at least one coordinate in N, or in other words, products with at least one factor in N. Thus U is a neighborhood of En−1 in En and p(U) is a neighborhood of Bn−1 in Bn. The homotop... |
−1(b)→p−1 w)v, which can be written F 1(b) wv ֏ Æ c F1( F1( F1( F1( c c c w homeomorphism, so this map is a homotopy equivalence, as desired. F 1(b) It remains to see that p is a quasifibration over Bn−Bn−1 and over the intersection of this set with U. The argument will be the same in both cases. c En − En−1, p−1(b) Ide... |
) this finishes the proof that SP (A)→SP (X)→SP (X/A) is a quasifibration. Since the homotopy axiom is obvious, this gives us the first two of the three axioms needed for the groups hi(X) = πiSP (X) to define a reduced homology theory. There remains only the wedge sum axiom, hi( α hi(Xα), but this is imα Xα) = mediate from... |
the Hurewicz homomorphism. For by definition of the Hurewicz homomorphism and naturality this reduces to the case X = S i, where the map SP1(S i) ֓ SP (S i) induces on πi a homomorphism Z→Z, which one just has to check is an isomorphism, the Hurewicz homomorphism being determined only up to sign. The suspension isomorp... |
) The restriction of p over each path-component of B is a quasifibration according to the definition in this section. Show these three conditions are equivalent, and prove Lemma 4K.3 for quasifibrations over non-pathconnected base spaces. 6. Let X be a complex of spaces over a complex, as defined in §4.G. Show that the nat... |
constructing the squares and powers and showing they satisfy the basic properties listed in the first part. More extensive 488 Chapter 4 Homotopy Theory applications will be given in [SSAT] after spectral sequences have been introduced. Most applications of Steenrod squares and powers do not depend on how these op- era... |
set of : Hm(X; G)→H n(X; H) and H n(K(G, m); H), defined all cohomology operations by ֏ (ι) where ι ∈ Hm(K(G, m); G) is a fundamental class. Θ Θ Θ Proof: Via CW approximations to spaces, it suffices to restrict attention to CW complexes, so we can identify Hm(X; G) with hX, K(G, m)i when m > 0 by Theorem 4.57, and with [... |
: H n(X; Z2)→H n+i(X; Z2), i ≥ 0, will satisfy the fol- lowing list of properties, beginning with naturality: (1) Sqi(f ∗(α)) = f ∗(Sqi(α)) for f : X→Y. (2) Sqi(α + β) = Sqi(α) + Sqi(β). (3) Sqi(α ` β) = (4) Sqi(σ (α)) = σ (Sqi(α)) where σ : H n(X; Z2)→H n+1( j Sqj (α) ` Sqi−j (β) (the Cartan formula). P X; Z2) is the... |
ial. n−1 is nonzero, Proof: Associated to a map f : S ℓ→Sm is the mapping cone Cf = Sm ∪f eℓ+1 with the cell eℓ+1 attached via f. Assuming f is basepoint-preserving, we have the relation C denotes reduced suspension. Σ f = Cf where Σ Σ If f : S 2n−1→S n has Hopf invariant 1, then by (5), Sqn : H n(Cf ; Z2)→H 2n(Cf ; Z2... |
i(α) = αp in (5) can only be expected to hold for evendimensional classes α since for odd-dimensional α the commutativity property of cup product implies that α2 = 0 with Zp coefficients if p is odd, and then αp = 0 since α2 = 0. Note that the formula P i(α) = αp for |α| = 2i implies that P i raises dimension by 2i(p − ... |
The Cartan formulas then say that Sq(α ` β) = Sq(α) ` Sq(β) and P (α ` β) = P (α) ` P (β), so Sq and P are ring homomorphisms. We can use Sq and P to compute the operations Sqi and P i for projective spaces and lens spaces via the following general formulas: (∗) Sqi(αn) = P i(αn) = n i αn+i for α ∈ H 1(X; Z2) αn+i(p−1... |
. Example 4L.3: Stable Splittings. The formula (∗) tells us how to compute Steenrod squares for RP∞, hence also for any suspension of RP∞. The explicit formu) and Sq(α2k−1) above show that all the powers of the generator las for Sq(α2k α ∈ H 1(RP∞; Z2) are tied together by Steenrod squares since the first formula connec... |
we have P i(αn) = dimension by 4i when p = 3, there is no chance that all the even-dimensional cohomology will be connected by the P i ’s. In fact, we showed in Proposition 4I.3 that K(Z3, 1) in dimensions congruent to 2 and 3 mod 4, while X2 has the remaining cohomology. Thus the best one Σ could hope would be that a... |
exercises. Σ Example 4L.4: Maps of HP∞. We can use the operations P i together with a bit of number theory to demonstrate an interesting distinction between HP∞ and CP∞, namely, we will show that if a map f : HP∞→HP∞ has f ∗(γ) = dγ for γ a generator of H 4(HP∞; Z), then the integer d, which we call the degree of f, m... |
(γ) = f ∗(2γ(p+1)/2) = 2d(p+1)/2γ(p+1)/2 and P 1f ∗(γ) = P 1(dγ) = 2dγ(p+1)/2 Hence the degree d satisfies d(p+1)/2 ≡ d mod p for all odd primes p. Thus either d ≡ 0 mod p or d(p−1)/2 ≡ 1 mod p. In both cases d is a square mod p since the Steenrod Squares and Powers Section 4.L 493 congruence d(p−1)/2 ≡ 1 mod p is equiv... |
odd qi. By the Chinese remainder theorem, this means specifying p modulo 8 times a product of odd primes. Dirichlet’s theorem guarantees p q that in fact infinitely many primes p exist satisfying this congruence condition. It is known that the integers realizable as degrees of maps HP∞→HP∞ are exactly the odd squares a... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.