text stringlengths 270 6.81k |
|---|
0 and T∞ where the radii r0 and r1 are zero, making the tori T0 and T∞ degenerate to circles. These two circles are the unit circles in the two C factors of C2, so under stereographic projection of S 3 from the point (0, 1) onto R3 they correspond to the unit circle in the xy plane and the z axis. The concentric tori T... |
�ber bundle with fiber S 7, the unit octonions. Let U0 and U1 be the complements of ∞ and 0 in the base space O ∪ {∞}. Define hi : p−1(Ui)→Ui × S 7 and gi : Ui × S 7→p−1(Ui) by h0(z0, z1) = (z0z−1 h1(z0, z1) = (z0z−1 1, z1/|z1|), 1, z0/|z0|), g0(z, w) = (zw, w)/|(zw, w)| g1(z, w) = (w, z−1w)/|(w, z−1w)| Elementary Method... |
Hopf map S 15→S 8, just as CP2 and HP2 are obtained from the other Hopf maps. However, there is no octonion analog of RPn, CPn, and HPn for n > 2 since associativity of multiplication is needed for the relation (z0, ···, zn) ∼ λ(z0, ···, zn) to be an equivalence relation. There are no fiber bundles with fiber, total spa... |
(In × {0} ∪ ∂In × I) ⊂ p−1(Uα), and composing the subcubes C. To extend this e e e e 380 Chapter 4 Homotopy Theory hα reduces us to the case of a product bundle Uα × F. In this case the first coordinate gt is just the given gt, so only the second coordinate needs to be constructed. of a lift This can be obtained as a co... |
3) ≈ πn(S 2) for all n ≥ 3. Taking n = 3, we see that π3(S 2) is infinite cyclic, generated by the Hopf map S 3→S 2. From this example and the preceding one we see that S 2 and S 3 × CP∞ are simplyconnected CW complexes with isomorphic homotopy groups, though they are not homotopy equivalent since they have quite differ... |
Hopf maps S 3→S 2 α S 2 W α) ⊕ L Q α π3(S 2 The factor ample. For the other factor we have π4( tion 4.28. The quotient L α S 2 α/ α S 2 α, α S 2 α has 5 skeleton a wedge of spheres S 4 W α S 2 W α) ≈ π4( Q Q α by the preceding exα S 2 α) by Proposiα/ αβ for α ≠ β, α S 2 W Q Elementary Methods of Calculation Section 4.... |
otopies of [f, g], we have a well-defined product πk(X)× πℓ(X)→πk+ℓ−1(X). The notation [f, g] is used since for k = ℓ = 1 this is just the commutator product in π1(X). It is an exercise to show that when k = 1 and ℓ > 1, [f, g] is the difference between g and its image under the π1 action of f. In these terms the map S 3... |
�) Taking the real case first, the Stiefel manifold Vn(Rk) is the space of n frames in Rk, that is, n tuples of orthonormal vectors in Rk. This is topologized as a subspace of the product of n copies of the unit sphere in Rk. The Grassmann manifold Gn(Rk) is the space of n dimensional vector subspaces of Rk. There is a ... |
, then applying the Gram–Schmidt process to this basis to make it orthonormal. The formulas for the Gram–Schmidt process show that it is continuous. Having orthonormal bases for all n planes in U, we can use these to identify these n planes with Rn, hence n frames in these n planes are identified with n frames in Rn, an... |
independence, so after again applying Gram–Schmidt we have a deformation through n frames, which finishes the construction of a contraction of Vn(R∞). Since Vn(R∞) is contractible, we obtain isomorphisms πiO(n) ≈ πi+1Gn(R∞) for all i and n, and similarly in the complex and quaternionic cases. Elementary Methods of Calc... |
k − m) is regarded as the subgroup of O(k) fixing the first m standard basis vectors. So we see that Vm(Rk) is identifiable with the coset space O(k)/O(k − m), or in other words the orbit space for the free action of O(k −m) on O(k) by right-multiplication. In similar fashion one can see that Gm(Rk) is the coset space O(k... |
, and the same is true for the groups πiU(n) and πiSp(n) via the other two bundles. One of the most surprising results in all of algebraic topology is the Bott Periodicity Theorem which asserts that these stable groups repeat periodically, with a period of eight for O and Sp and a period of two for U. Their values are ... |
of order a power of p, but the i Elementary Methods of Calculation Section 4.2 385 The figure below is a schematic diagram of the 2 components of π s i for i ≤ 60. A vertical chain of n dots in the ith column represents a Z2n summand of π s i. The bottom dot of such a chain denotes a generator of this summand, and the ... |
� Z8 is a quotient of π s 3. Remember that 2π s Across the bottom of the diagram there is a repeated pattern of pairs of ‘teeth’. This pattern continues to infinity, though with the spikes in dimensions 8k − 1 not all of the same height, namely, the spike in dimension 2m(2n + 1) − 1 has height m + 1. i for i ≤ 100. Here... |
these dimensions. In the diagram of 2π s ∗ these are the parts of the teeth connected to the spike in dimension 8k − 1. The J homomorphism will be studied in some detail in [VBKT]. ∗ include classes ηn ∈ 2π s 2n for n ≥ 4, βn ∈ pπ s 2(p3−1)n−2p2−2p+1 for p ≥ 7. The element βn appears in the diagram for p = 5 as the do... |
suspension SW as the union of two cones CW, define the Toda bracket hf, g, hi : SW→Z to be the composition G(Cf ) on one cone and hF on the other. f-----→ X g-----→ Y The map hf, g, hi is not uniquely determined by f, g, and h since it depends on the choices of the nullhomotopies. In the case of π s ∗, the various choi... |
smash product S n ∧ S 1 can be regarded as the quotient space of S n × I with S n × ∂I ∪ {x0}× I collapsed to a point. This is the same as the quotient of the suspension S n+1 of S n obtained by collapsing to a point the suspension of x0. Collapsing this arc in S n+1 to a point again yields S n+1, so we obtain in this... |
)(j+k), which equals (−1)ij since k is even. The composition (g ∧ f )σ is homotopic to (−1)ij (g∧f ) since additive inverses in homotopy groups are obtained by precomposing with a reflection, of degree −1. Thus from the commutativity of the diagram we obtain the relation f ∧ g ≃ (−1)ij g ∧ f. ⊔⊓ Elementary Methods of Ca... |
Let X be the mapping torus of f, the quotient space of (S 2 α ∨ S 2 6. Show that the relative form of the Hurewicz theorem in dimension n implies the absolute form in dimension n − 1 by considering the pair (CX, X) where CX is the cone on X. 7. Construct a CW complex X with prescribed homotopy groups πi(X) and prescri... |
otopy Theory homology with local coefficients as saying that f∗ : H∗(X; Z[π1X])→H∗(Y ; Z[π1Y ]) is an isomorphism; see §3.H.] 13. Show that a map between connected n dimensional CW complexes is a homotopy equivalence if it induces an isomorphism on πi for i ≤ n. [Pass to universal covers and use homology.] 14. If an n di... |
Y is contractible. [Use the K¨unneth formula.] 19. If X is a K(G, 1) CW complex, show that πn(X n) is free abelian for n ≥ 2. 20. Let G be a group and X a simply-connected space. Show that for the product K(G, 1)× X the action of π1 on πn is trivial for all n > 1. Hi(X) and e e 21. Given a sequence of CW complexes K(G... |
The previous exercise is needed for the case n > 1.] π2(X) ≈ G iff H2(K(G, 1); Z) = 0. 24. Show there is a Moore space M(G, 1) with π1 [Use the preceding problem. Build such an M(G, 1) from the 2 skeleton K2 of a K(G, 1) by attaching 3 cells according to a basis for the free group H2(K2; Z).] In particular, there is no... |
��′ 29. Finish the homotopy classification of lens spaces begun in Exercise 2 of §3.E by showing that two lens spaces Lm(ℓ1, ···, ℓn) and Lm(ℓ′ n) are homotopy equivalent if ℓ1 ··· ℓn ≡ ±knℓ′ n mod m for some integer k, via the following steps: (a) Reduce to the case k = 1 by showing that Lm(ℓ′ n) if k is relatively pri... |
R, (x, y) ֏ x, a fiber bundle? 31. For a fiber bundle F→E→B such that the inclusion F ֓ E is homotopic to a constant map, show that the long exact sequence of homotopy groups breaks up into split short exact sequences giving isomorphisms πn(B) ≈ πn(E) ⊕ πn−1(F ). In particular, for the Hopf bundles S 3→S 7→S 4 and S 7→S ... |
, where f g denotes the action of f on g. e 37. Show that all Whitehead products in a path-connected H–space are trivial. 38. Show π3(S 1 ∨S 2) is not finitely generated as a module over Z[π1(S 1 ∨S 2)] by considering Whitehead products in the universal cover, using the results in Example 4.52. Generalize this to πi+j−1... |
ilenberg–MacLane spaces, up to homotopy equivalence. The most ge- ometric interpretation of the phrase ‘twisted product’ is the notion of fiber bundle introduced in the previous section, but here we need the more homotopy-theoretic notion of a fibration, so before we begin the discussion of Postnikov towers we first take ... |
K; Z), G) given by the inverse of the Hurewicz isomorphism G = πn(K(G, n))→Hn(K; Z). Concretely, if we choose K(G, n) to be a CW complex with (n − 1) skeleton a point, then a fundamental class is represented by the cellular cochain assigning to each n cell of K(G, n) the element of πn(K(G, n)) defined by a characteristi... |
natural isomorphisms hn(X) ≈ H n(X; h0(S 0)) for all CW complexes X and all n. e Towards proving (1) we will study a more general question: When does a sequence of spaces Kn define a cohomology theory by setting hn(X) = hX, Kni? Note that this will be a reduced cohomology theory since hX, Kni is trivial when X is a poi... |
, Ki. The space X Σ Σ suspension of X when we want to distinguish it from the ordinary suspension SX. Σ Σ It is easy to check that h X, Ki is a group with respect to the sum defined above, inverses being obtained by reflecting the I coordinate in the suspension. However, what we would really like to have is a group struc... |
X ≃ Y implies Ω Ω X ≃ Y. homotopy type of a CW complex. This may be a bit surprising since loopspaces are Ω Ω Ω Ω usually quite large spaces, though of course CW complexes can be quite large too, in terms of the number of cells. What often happens in practice is that if a CW complex X has only finitely many cells in ea... |
= hX Thus for a sequence of spaces Kn to define a cohomology theory hn(X) = hX, Kni we have been led to the assumption that each Kn should be a loopspace and in fact a double loopspace. Actually we do not need Kn to be literally a loopspace since it would suffice for it to be homotopy equivalent to a loopspace, as hX, Kn... |
nor mentioned above it would be possible to X, Kn+1i = hX, Kn+1→ Ω K(G, n + 1 replace ‘weak homotopy equivalence’ by ‘homotopy equivalence’ in this definition. However it does not noticeably simplify matters to do this, except perhaps psycho- logically. Notice that if we discard a finite number of spaces Kn from the begi... |
K–theory. For a more in-depth introduction to the theory of infinite loopspaces, the book [Adams 1978] can be much recommended. Ω Ω Proof: Two of the three axioms for a cohomology theory, the homotopy axiom and the wedge sum axiom, are quite easy to check. For the homotopy axiom, a basepointpreserving map f : X→Y induc... |
suspensions, and so on. The resulting infinite sequence can be written in either of the following two forms: A→X→X ∪ CA→SA→SX→S(X ∪ CA)→S 2A→S 2X→ ··· A→X→X/A→SA→SX→SX/SA→S 2A→S 2X→ ··· In the first version we use the obvious equality SX ∪ CSA = S(X ∪ CA). The first version has the advantage that the map X ∪CA→SA is easi... |
) hA, Ki← hX, Ki← hX/A, Ki← h A, Ki← h X, Ki← ··· whose maps are defined by composition with those in (2). For example, the map hX, Ki→hA, Ki sends a map X→K to A→X→K. The sets in (3) are groups starting Σ Σ Connections with Cohomology Section 4.3 399 Σ with h A, Ki, and abelian groups from h 2A, Ki onward. It is easy t... |
, A)→(Y, B) since cofibration sequences ⊔⊓ are natural. There is no essential difference between cohomology theories on basepointed CW complexes and cohomology theories on nonbasepointed CW complexes. Given a h∗, one gets an unreduced theory by setting reduced basepointed cohomology theory hn(X/A), where X/∅ = X+, the un... |
homology groups hn(X) by the same argument as for ordinary homology. The main thing to verify now is that this cellular chain complex is isomorphic to the cellular chain complex in ordinary homology with coefficients in the group G = h0(point ). Certainly the cellular chain groups in the two cases are isomorphic, being ... |
inclusions X. Let X -→ q1, q2 : Σ maps restricting to the identity on the Σ X ֓ X be the quotient summand indicated by the subscript and collapsing the other summand to a point. Then q1∗ ⊕ q2∗ is an inverse to i1∗ ⊕ i2∗ since qj ik is the identity map for j = k and the constant map for An element x in the left-hand gr... |
the identity, and similarly (0, y) maps to y. Hence (x, y) maps to x + y in the left-hand group. We conclude that u ∈ h(K) maps by the composition across the top of the diagram to f ∗(u) + g∗(u) in h( (f + g)∗ by definition. X). But this composition is ⊔⊓ Σ Σ Σ Returning to the proof of the theorem, we see that the cel... |
the cellular coboundary maps are uniquely determined by how they map factors of one direct product to factors of the other direct product. To be precise, consider the cellular coboundary map dn : hn(X n, X n−1)→hn+1(X n+1, X n). Decomposing the latter group as a product of copies of G for the (n + 1) cells, we see tha... |
G, n); G), independent of f. This is purely formal: Take α = T (11) for 11 the identity map of K(G, n), and then naturality gives T ([f ]) = T (f ∗(11)) = f ∗T (11) = f ∗(α), where the first f ∗ refers to induced homomorphisms for the functor hn, which means composition ⊔⊓ with f. e e The fundamental class α = T (11) ca... |
hX, K(G, n)i leads to a basic principle which reappears many places in algebraic topology, the idea that the occurrence or nonoccurrence of a certain phenomenon is governed by what happens in a single spe- cial case, the universal example. To illustrate, let us prove the following special fact: Connections with Cohomo... |
�(K(Z, n); Z) generated by the fundamental class α for odd n ≥ 3. By the commutativity property of cup products we know that α2 is either zero or of order two. To see that α2 is nonzero it suffices to find a single space X with an element γ ∈ H n(X; Z) such that γ2 ≠ 0. The first place to look might be RP∞, but its cohomol... |
section. However, for larger n it quickly becomes impractical to make this procedure explicit since homotopy groups are so hard to compute. One can get some idea of the difficulties of the next case n = 3 by considering the homology groups of K(Z, 3). Using techniques in [SSAT], the groups Hi(K(Z, 3); Z) for 0 ≤ i ≤ 12 ... |
G, n)i. Taking Z coefficients for simplicity, an element of Hm(K(Z, n); Z) corresponds to a map θ : K(Z, n)→K(Z, m). We can compose θ with any map f : X→K(Z, n) to get a map θf : X→K(Z, m). Letting f vary and keeping θ fixed, this gives a function H n(X; Z)→Hm(X; Z), depending only on θ. This is the idea of cohomology ope... |
take Sm as the (m + 1) skeleton of Km, and similarly for Kn, so Km ∧ Kn has Sm ∧ S n as its (m + n + 1) skeleton and we can obtain µ by extending the inclusion Sm ∧ S n = Sm+n ֓ Km+n. It is not hard to prove the basic properties of cup product using this definition, and in particular the commutativity property becomes ... |
for all t. In particular, gt : Fγ(0)→E with e e e pendent of the choice of the lifting gt of gt. (b) For a composition of paths γγ′, Lγγ′ is homotopic to the composition Lγ′ Lγ. e From these statements it follows that Lγ is a homotopy equivalence with homotopy inverse Lγ, where γ is the inverse path of γ. Before provi... |
4 Homotopy Theory p : p−1(A)→A is a fibration for any subspace A ⊂ B. So we can ask whether every point of B has a neighborhood U for which the fibration p−1(U)→U is equivalent in some homotopy-theoretic sense to a projection U × F→U. The natural notion of equivalence for fibrations is defined in the following way. Given ... |
homotopy gt : X→A gives the first gt : X→f ∗(E), the second coordinate being coordinate of a lift a lifting to E of the composed homotopy f gt. e Proposition 4.62. Given a fibration p : E→B and a homotopy ft : A→B, the pullback fibrations f ∗ 1 (E)→A are fiber homotopy equivalent. 0 (E)→A and f ∗ Proof: Let F : A× I→B be ... |
homotopy equivalent to a product fibration B × F→B. Connections with Cohomology Section 4.3 407 Proof: The pullback of E by the identity map B→B is E itself, while the pullback by a constant map B→B is a product B × F. ⊔⊓ Thus we see that if B is locally contractible then any fibration over B is locally fiber homotopy eq... |
A× BI and then apply (b) of Proposition A.14 which in the current context asserts that continuity of a map X × I→A× BI is equivalent to continuity of the associated map X × I × I→A× B. ⊔⊓ g0(x) = γx(1). To check that e e e e e We can regard A as the subspace of Ef consisting of pairs (a, γ) with γ the constant path at... |
P B of paths in B starting at b0, and p : P B→B sends each path to its endpoint. The fiber p−1(b0) is the loopspace B consisting of all loops in B based at b0. Since P B is contractible by progressively truncating paths, the long exact sequence of homotopy groups for the path fibration P B→B yields another proof that πn... |
. We have h0 = 11, h1(Ep) ⊂ E, and ht(E) ⊂ E for all t. If we let i denote the inclusion E ֓ Ep, then ih1 ≃ 11 via ht and h1i ≃ 11 via ht || E, so i is a ⊔⊓ fiber homotopy equivalence. e e e We have seen that loopspaces occur as fibers of fibrations P B→B with con- tractible total space P B. Here is something of a convers... |
hold in general: For each topological group G there is a fiber bundle G→EG→BG with EG contractible, hence by the proposition there is a weak equivalence G ≃ BG. There is also a converse statement: The loopspace Ω Ω of a CW complex is homotopy equivalent to a topological group. Ω Ω The relationship between X and X has b... |
The homotopy fiber Fi consists of pairs (γ, η) where η is a path in E ending at e0 and γ is a path in B from p(η(0)) to b0. A homotopy inverse to the B sending (γ, η) to the loop obtained by inclusion B, and the inclusion Ω Ω B ֓ Fi is the retraction Fi→ composing the inverse path of pη with γ. Ω Ω These constructions ... |
composition Xn→Xn−1 ֓ X ′ n−1 fitting into the commutative diagram at the right. Thus we obtain a n−1 into a fibration X ′ n→X ′ n→X ′ n−1 in succession, starting Postnikov tower satisfying also the condition (3) The map Xn→Xn−1 is a fibration with fiber a K(πnX, n). To the extent that fibrations can be regarded as twisted... |
) by maps fn : (S i, s0)→(Xn, xn). Since the projection pn : Xn→Xn−1 takes [fn] to [fn−1], by applying the homotopy lifting Connections with Cohomology Section 4.3 411 property for the pair (S i, s0) we can homotope fn, fixing s0, so that pnfn = fn−1. Doing this inductively for n = 2, 3, ···, we get pnfn = fn−1 for all ... |
πi+1(Xn), where lim is not relevant. ←-is the functor defined in §3.F. Namely, if f : S i→ lim ←-- Xn determines an element of Ker λ, then the sequence of maps gn : S i+1→Xn constructed above gives an element n πi+1(Xn), well-defined up to the choice of the nullhomotopies Fn. Any new of choice of Fn is obtained by adding... |
would be very nice if the fibration K(π, n)→Xn→Xn−1 could be extended another term to the right, to form a fibration sequence K(π, n)→Xn→Xn−1→K(π, n + 1) 412 Chapter 4 Homotopy Theory for this would say that Xn is the homotopy fiber of a map Xn−1→K(π, n + 1), and homotopy classes of such maps are in one-to-one correspond... |
and in the general case X is some sort of twisted product of K(πnX, n) ’s. To actually build a space from its k invariants is usually too unwieldy a procedure to be carried out in practice, but as a theoretical tool this procedure can be quite useful. The next result tells us when this tool is available: Theorem 4.69.... |
the action of π1(F ) on πn(E, F ) being trivial, which is always the case in a fibration since under the isomorphism p∗ : πn(E, F )→πn(B, x0) an element γα−α, with γ ∈ π1(F ) and α ∈ πn(E, F ), maps to p∗(γ)p∗(α) − p∗(α) which is zero since p∗(γ) lies in the trivial group π1(x0). The relative group πn(X, A) is always i... |
urewicz theorem gives πn+1(X/A) ≈ Hn+1(X/A). Hence the quotient map X→X/A induces an isomorphism πn+1(X, A) ≈ πn+1(X/A) since the analogous statement for homology is certainly true. Since πn+1(X/A) ≈ π, we can build a K(π, n + 1) from X/A by attaching cells of dimension n + 3 and greater. This leads to the commutative ... |
A Moore–Postnikov tower for f is a commutative diagram as shown at the right, with each composition X→Zn→Y homotopic to f, and such that: (1) The map X→Zn induces an isomorphism on πi for i < n and a surjection for i = n. (2) The map Zn→Y induces an isomorphism on πi for i > n and an injection for i = n. (3) The map Z... |
→Zn→Y are inclusions by taking mapping cylinders, first of X→Zn+1, then of the new Zn+1→Zn, and then of the new Zn→Y. From the left-hand triangle we see that Zn+1→Zn induces an isomorphism on πi for i < n and a surjection for i = n, hence πi(Zn, Zn+1) = 0 for i < n + 1. Similarly, the other triangle gives πi(Zn, Zn+1) =... |
Zn−1→K(πnY, n) that is the first nontrivial stage in a Postnikov tower for Zn−1. A generalization of the preceding theory allowing nontrivial actions of π1 can be found in [Robinson 1972]. Obstruction Theory It is very common in algebraic topology to encounter situations where one would like to extend or lift a given m... |
skeleta of W. This approach has an appealing directness, but the technical details of working at the level of cochains are perhaps a little tedious. Instead of pursuing this direct line we shall follow the second approach, which is slightly more sophisticated but has the advantage that the theory becomes an almost tri... |
. The map W→K together with the nullhomotopy on A gives a map W ∪ CA→K, where CA is the cone on A. Since K is a K(πnX, n + 1), the map W ∪ CA→K determines an obstruction class ωn ∈ H n+1(W ∪ CA; πnX) ≈ H n+1(W, A; πnX). Proposition 4.72. A lift W→Xn extending the given A→Xn exists iff ωn = 0. Proof: We need to show that... |
i, so by Lemma 4.6, the compression lemma, the map (W, A)→(M, X) can be homotoped to a map W→X extending the given A→X, and we have solved the extension problem. Thus if it happens that at each stage of the inductive process of constructing lifts W→Xn the obstruction ωn ∈ H n+1(W, A; πnX) vanishes, then the extension ... |
on homology, we have H∗(Y, X) = 0, hence H n+1(Y, X; πn(X)) = 0 for all n by the universal coefficient theorem. So there are no obstructions, and a retraction Y →X exists. This implies that the maps πn(Y )→πn(Y, X) are onto, so trivial action of π1(X) on πn(Y ) implies ⊔⊓ trivial action on πn(Y, X) by naturality of the ... |
be many choices of such a lift, and different choices could lead to different ωn+1 ’s, some zero and others nonzero. Examples of such ambiguities are not hard to produce, for both the lifting and the extension problems, and the Connections with Cohomology Section 4.3 419 ambiguities only become worse with each subsequen... |
. K(Z, n) from S n by attaching cells of dimension ≥ n + 2.] 6. Use Exercise 4 to construct a multiplication map µ : K(G, n)× K(G, n)→K(G, n) for any abelian group G, making a CW complex K(G, n) into an H–space whose multipli- [Build a cation is commutative and associative up to homotopy and has a homotopy inverse. Sho... |
B1→B2 induces f ∗ : F(B2)→F(B1) depending only on the homotopy class of f, with f ∗ a bijection if f is a homotopy equivalence. 12. Show that for homotopic maps f, g : A→B the fibrations Ef →B and Eg→B are fiber homotopy equivalent. 13. Given a map f : A→B and a homotopy equivalence g : C→A, show that the fibrations Ef →... |
exact sequence in the preceding problem can be improved to the statement that two elements of hX, F i have the same image in hX, Ei iff they are in the same orbit of the induced action of p-----→ B, define a natural action of Ω Ω Ω e hX, Bi on hX, F i. 20. Show that by applying the loopspace functor to a Postnikov tower... |
the set [Z, X] of unrestricted homotopy classes of maps Z→X, for Z any CW complex with basepoint z0 a 0 cell. Then the section concludes with an extended example exhibiting some rather subtle nonfinite generation phenomena in homotopy and homology groups. We begin by constructing an action of π1(X, x0) on hZ, Xi when Z... |
epoint-preserving homotopy from f0 to g0 on Z × {0}× I, and the homotopy from fs(z0) to gs(z0) on {z0}× I × I. We would like to extend H over Z × I × I. The pair (I × I, I × ∂I ∪ {0}× I) is homeo- [f0] = [η] [f0][γ] [γ][η] [γ][η] [η]. = morphic to (I × I, I × {0}), and via this homeomorphism we can view H as a map Z × ... |
�f in terms of maps (In, ∂In)→(X, x0), a homotopy from γf to f is obtained by restricting γf to smaller and smaller concentric cubes, and on the ‘basepoint’ ∂In this homotopy traces out the loop γ. Proposition 4A.2. If (Z, z0) is a CW pair and X is a path-connected space, then the natural map hZ, Xi→[Z, X] induces a bi... |
n. For example, when n = 1 the orbits are just the conjugacy classes in π1, and these form a group only when π1 is abelian. Basepoints are thus a necessary technical device for producing the group structure in homotopy groups, though as we have shown, they can be ignored in simply-connected spaces. For a set of maps S ... |
the 2n dimensional torus T 2n, the product of 2n circles. Define f : T 2n→S 1 by f (θ1, ···, θ2n) = θ1 + ··· + θ2n where the coordinates θi ∈ S 1 are viewed as angles measured in radians. The space Z = X ∩ f −1(0) will provide the example we are looking for. As we shall see, Z is a finite CW complex of dimension n − 1, ... |
Z)→Hn−1(Z) the first term is zero and the last term is finitely generated, Z being a finite CW complex of dimension n − 1, 424 Chapter 4 Homotopy Theory while the third term is an infinite sum of Z ’s, one for each n cell of Y. If πn−1(Z) were finitely generated as a π1(Z) module, then by attaching finitely many n cells to ... |
a cone are also cones, of lower dimension. The slabs, together with all their lower-dimensional faces, give a CW structure on Rm with the planes of L−1(Z) as subcomplexes. These structures are preserved by the deck transformations of the cover Rm→T m so there is an induced CW structure in the quotient T m, with f −1(0... |
homotopy e equivalence gi : Yi→ X[−i, i] have already been defined, we form Yi+1 by attaching two n cells by the maps obtained from the attaching maps of the two n cells in X[−i, i] by composing with a homotopy inverse to gi. This allows X[−i − 1, i + 1]. Taking the X[−i − 1, i + 1] − gi to be extended to a homotopy eq... |
regular octahedron inscribed in the cube with vertices (±1, ±1, ±1). If we identify each pair of oppo- site edges of the octahedron, each pair of opposite triangular faces becomes a torus. However, there are only four pairs of opposite faces, so we get only four tori this way, not eight. To correct this problem, regar... |
. 2. Show that under the map hX, Y i→Hom, [f ] ֏ f∗, the acπn(X, x0), πn(Y, y0) tion of π1(Y, y0) on hX, Y i corresponds to composing with the action on πn(Y, y0), that is, (γf )∗ = βγf∗. Deduce a bijection of [X, K(π, 1)] with the set of orbits of Hom(π1(X), π ) under composition with inner automorphisms of π. In part... |
ber bundle (c) Use part (b) to find a presentation for π1( Xn), and show this presentation reduces e to a finite presentation if n > 2 and a presentation with a finite number of generators e if n = 2. In the latter case, deduce that π1( X2) has no finite presentation from the fact that H2( X2) is not finitely generated. Xn−... |
H(f ) depends on the choice of the gener- ator β, but this can be specified by requiring β to correspond to a fixed generator of H 2n(D2n, ∂D2n) under the map H 2n(Cf ) ≈ H 2n(Cf, S n)→H 2n(D2n, ∂D2n) induced by the characteristic map of the cell e2n, which is determined by f. We can then change the sign of H(f ) by com... |
n = 2, 4, 8. This has a number of very interesting con- sequences, for example: 428 Chapter 4 Homotopy Theory Rn is a division algebra only for n = 1, 2, 4, 8. S n is an H–space only for n = 0, 1, 3, 7. S n has n linearly independent tangent vector fields only for n = 0, 1, 3, 7. The only fiber bundles S p→S q→S r occur... |
induced cellular chain map q∗ sends e2n f +g to e2n g. In cohomology this implies that q∗(βf ) = q∗(βg) = βf +g where βf, βg, and βf +g are the cohomology classes dual to the 2n cells. Letting αf +g and αf ∨g be the cohomology classes corresponding to the n cells, we have q∗(αf ∨g) = αf +g since q is a homeomorphism o... |
generated homology groups is always homotopy equivalent to a CW complex having the minimum number of cells consistent with its homology, namely, one n cell for each Z summand of Hn and a pair of cells of dimension n and n + 1 for each Zk summand of Hn. Proposition 4C.1. Given a simply-connected CW complex X and a deco... |
do not contribute to Hn(Z n). Thus Hn(Z n) is free with basis the generator n cells, and the kernel of Hn(Z n)→Hn(X) is free with basis given by certain multiples of some of the generator n cells. Choose ‘relator’ elements ρi in Hn+1(Mf, Z n) mapping to this basis for the kernel, and let the ‘generator’ elements γi ∈ ... |
In particular, f∗ : Hn+1(Z n+1)→Hn+1(X) is surjective, and the induction step is finished. Doing this for all n, we produce a CW complex Z and a map f : Z→X with the ⊔⊓ desired properties. Example 4C.2. Suppose X is a simply-connected CW complex such that for some n ≥ 2, the only nonzero reduced homology groups of X ar... |
the Whitehead products [ij, ik], j < k, where ij is the inclusion S 2 j. Since W a homotopy of ϕ does not change the homotopy type of Xϕ, we may assume ϕ is j<k ajk[ij, ik]. We need to see how the coefficients a linear combination aj and ajk determine the cup product H 2(X; Z)× H 2(X; Z)→H 4(X; Z). j S 2 W j ajηj + j ֓ ... |
induced map q∗ : H 4(Xϕ,ψ)→H 4(Xϕ+ψ) sends each of the two generators corresponding to the 4 cells of Xϕ,ψ to a generator, and the assertion follows. Now suppose Xϕ and Xψ have isomorphic cup product rings. This means bases for H ∗(Xϕ; Z) and H ∗(Xψ; Z) can be chosen so that the matrices specifying the cup product H 2... |
ffice. As applications we calculate the cohomology rings of some important spaces closely related to Lie groups. In particular we find a number of spaces with exterior and polynomial cohomology rings. The Leray–Hirsch Theorem This theorem will be the basis for all the other results in this section. It gives hypotheses suffi... |
irsch theorem does not assert that the isomorphism H ∗(E; R) ≈ H ∗(B; R) ⊗R H ∗(F ; R) is a ring isomorphism, and in fact this need not be true, for example for the Klein bottle viewed as a bundle with fiber and base S 1, where the Leray–Hirsch theorem applies with Z2 coefficients. An example of a bundle where the classes... |
(b)` cj = p∗(δb)` cj since δcj = 0. Φ The space B′ deformation retracts onto the skeleton Bn−1, and the following lemma implies that the inclusion p−1(Bn−1) ֓ E′ is a weak homotopy equivalence, hence induces an isomorphism on all cohomology groups: Lemma 4D.2. Given a fiber bundle p : E→B and a subspace A ⊂ B such that ... |
diagram with (B, B′) replaced by (U, U ′), induction implying that the theorem holds for U and U ′ since they deformation retract onto complexes of dimensions 0 and n − 1, respectively, and by the lemma we S Φ can restrict to the bundles over these complexes. Next there is the case that B is an infinite-dimensional CW ... |
�(f ∗(E); R) which still restrict to a basis in each fiber, and so the ⊔⊓ naturality of reduces the theorem for E→B to the case of f ∗(E)→A. Φ Corollary 4D.3. (a) H ∗(U(n); Z) ≈ generators xi of odd dimension i. (b) H ∗(SU(n); Z) ≈ (c) H ∗(Sp(n); Z) ≈ Z[x3, x5, ···, x2n−1]. Z[x3, x7, ···, x4n−1]. Λ Λ Z[x1, x3, ···, x2n−... |
−1; Z). By commutativity of cup product this is the exterior algebra Z[x1, ···, x2n−1]. Λ The same proof works for Sp(n) using the bundle Sp(n − 1)→Sp(n)→S 4n−1. In the case of SU(n) one uses the bundle SU(n − 1)→SU(n)→S 2n−1. Since SU(1) is the trivial group, the bundle SU(1)→SU(2)→S 3 shows that SU(2) = S 3, so the ⊔... |
generalizing the calculation of the cohomology rings of projective spaces: Theorem 4D.4. If Gn(C∞) is the Grassmann manifold of n dimensional vector subspaces of C∞, then H ∗(Gn(C∞); Z) is a polynomial ring Z[c1, ···, cn] on generators ci of dimension 2i. Similarly, H ∗(Gn(R∞); Z2) is a polynomial ring Z2[w1, ···, wn]... |
�flag’ is not exactly clear, but that is the traditional name. The set of all n flags in Ck forms a subspace Fn(Ck) of the product of n copies of CPk−1. There is a natural fiber bundle Fn(Cn) ------→ Fn(Ck) p------------→ Gn(Ck) where p sends an n tuple of orthogonal lines to the n plane it spans. The local triviality pro... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.