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→Vn,k such that pf = 11, is exactly a set of k − 1 orthonormal tangent vector fields v1, ···, vk−1 on S n−1 since f assigns to each x ∈ S n−1 an orthonormal k frame (x, v1(x), ···, vk−1(x)). We described a cell structure on Vn,k at the end of §3.D, and we claim that the (n − 1) skeleton of this cell structure is RPn−1/R... |
)→H n−1(RPn−1/RPn−k−1; Z2) will be nonzero, contradicting the existence of the map g since obviously the operation Sqk−1 : H n−k(S n−1; Z2)→H n−1(S n−1; Z2) is zero. In order to guarantee that n−k k−1 ≡ 1 mod 2, write n = 2r (2s + 1) and choose k = 2r + 1. Assume for the moment that s ≥ 1. Then, and in view of the rule... |
on the unit sphere in Hm. When r = 3 we can follow the same Steenrod Squares and Powers Section 4.L 495 procedure with the octonions, constructing seven orthonormal tangent vector fields to the unit sphere in Om via an orthonormal basis 1, i, j, k, ··· for O. The upper bound of 2r − 1 for the number of orthonormal vect... |
1 : H 3(X; Zp)→H 2p+1(X; Zp) is topy, a wedge summand of nontrivial. Since X is simply-connected, the construction in §4.C shows that we may take X to have (2p + 1) skeleton of the form S 2 ∪ e3 ∪ e2p ∪ e2p+1. In fact, using the notion of homology decomposition in §4.H, we can take this skeleton to be the reduced mapp... |
, n) has order 4 in the group of basepoint-preserving homotopy classes of maps M(Z2, n)→M(Z2, n), with addition defined via the suspension structure on M(Z2, n) = M(Z2, n − 1). According to Proposition 4H.2, this group is the middle term of a short exact sequence, the remaining terms of which contain only Σ 496 Chapter ... |
be a generator of H 1(RP2; Z2) and β a generator of H n−1(M(Z2, n − 1); Z2), we have Sq2α = 0 = Sq2β, but Sq1α and Sq1β are nonzero since Sq1 is the Bockstein. By the K¨unneth formula, Sq1α× Sq1β then generates H n+2(RP2 ∧ M(Z2, n − 1); Z2) and we are done. Adem Relations and the Steenrod Algebra When Steenrod squares... |
relations, that is, by the polynomials given by the differences between the left and right sides of the Adem relations. In similar fashion, A p for odd p is defined to be the algebra over Zp formed by polynomials in the noncommuting variables β, P 1, P 2, ··· modulo the Adem relations and the relation Steenrod Squares a... |
p + ··· + (p − 2)pk, and the p adic expansion of (p−1)b−1 a is i0 + i1p + ··· + (ik − 1)pk. Hence and in each factor a of the latter product the numerator is nonzero in Zp so the product is nonzero in Zp. When p = 2 the last factor is omitted, and the product is still nonzero in Z2. ⊔⊓ p−2 ik−1 p−1 i0 ··· ≡ element a o... |
= 2, Sqn(α) = α2 ≠ 0. If n is not a power of 2 then Sqn decomposes into compositions Sqn−jSqj with 0 < j < n. Such compositions must be zero since they pass through the group H n+j(X; Z2) which is zero for 0 < j < n. 498 Chapter 4 Homotopy Theory For odd p, the fact that α2 is nonzero implies that n is even, say n = 2... |
i(Cf ; Z2) must be trivial since it is a sum of compositions that pass through trivial cohomology groups, and similarly for P i. Interestingly enough, the Adem relations can also be used in a positive way to ∗, as the proof of the following result will show. detect elements of π s Proposition 4L.11. If η ∈ π s nonzero ... |
+4(X; Z2) is an isomorphism. But this is impossible in view of the Adem relation Sq2Sq2 = Sq3Sq1, since Sq1 is trivial on H n(X; Z2). Σ The same argument shows that ν 2 and σ 2 are nontrivial using the relations ⊔⊓ Sq4Sq4 = Sq7Sq1 + Sq6Sq2 and Sq8Sq8 = Sq15Sq1 + Sq14Sq2 + Sq12Sq4. This line of reasoning does not work f... |
all j, which again means that no Adem relation can be applied to the monomial. As with A 2, the Adem relations suffice to reduce every monomial to a linear combination of ad- missible monomials, by the same argument as before but now using the lexicographic ordering on tuples (ε1 + pi1, ε2 + pi2, ···). Define the excess ... |
q1ι, Sq4Sq2Sq1ι, ···] K(Zp, 2) : Zp[ι, βP 1βι, βP pP 1βι, βP p2 [ι]⊗Zp[βι] Λ Zp P pP 1βι, ···] ⊗ Zp [βι, P 1βι, P pP 1βι, P p2 P pP 1βι, ···] The theorem implies that the admissible monomials in A p are linearly indepenΛ dent, hence form a basis for A p as a vector space over Zp. For if some linear combination of admis... |
the one hand, this limit is exactly the stable opera- Σ tions by Proposition 4L.1 and the definition of a stable operation. On the other hand, the preceding theorem implies that this limit is A p since it says that all elements of H ∗(K(Zp, n); Zp) below dimension 2n are uniquely expressible as sums of admissible monom... |
), which is β, then P I (ιn) = β valid when |x| is even, as we may assume is the case here, otherwise (P J (ιn))pk = 0 by commutativity of cup product. SqJ (ιn) (P J (ιn))pk pk There is another set of relations among Steenrod squares equivalent to the Adem relations and somewhat easier to remember: k j Sq2n−k+j−1Sqn−j ... |
case p = 2 is in some ways simpler than the case p odd, so in the first part of the development we will specialize p to 2 whenever there is a significant difference between the two cases. 502 Chapter 4 Homotopy Theory Before giving the construction in detail, let us describe the idea in the case p = 2. The cup product sq... |
in K(Z2, 2n). This means that when we compose with the diagonal map S ∞ × X→S ∞ × X × X, (s, x)֏ (s, x, x), there is an induced quotient map RP∞ × X→K(Z2, 2n) extending α2 : X→K(Z2, 2n). This extended map represents a class in H 2n(RP∞ × X; Z2). By the K¨unneth formula and the fact that H ∗(RP∞; Z2) is the polynomial ... |
or RP∞ when p = 2. On the product S ∞ × X ∧p there is then the diagonal action g(s, x) = (g(s), g(x)) for X denote the orbit space (S ∞ × X ∧p)/Zp of this diagonal action. This g ∈ Zp. Let X ∧p described in §3.G. The projection is the same as the Borel construction S ∞ × Zp S ∞ × X ∧p→S ∞ induces a projection π : X→L∞... |
subcomplex, so the quotient quotient CW structure on X. The section L∞ ⊂ Γ Γ X inherits a CW structure from Γ X. In particular, note that if the n skeleton of X X is S pn with its usual Γ is S n with its usual CW structure, then the pn skeleton of Λ CW structure. Γ, 1,, and 1 are functors: A map f : (X, x0)→(Y, y0) Λ ... |
e X ∧p ⊂ Kn). X. By naturality it will suffice to construct λ(ι) ∈ H pn( e e e e e Λ The key point in the construction of λ(ι) is the fact that T ∗(ι⊗p) = ι⊗p. In terms Λ n →Kpn, this says the composition ι⊗p T is homotopic to ι⊗p, preserving of maps K∧p basepoints. Such a homotopy can be constructed as follows. The pn ... |
Kn by attaching cells of dimension greater than than n, Λ Kn→Kpn since pn + 1. There are then no obstructions to extending λ1 to a map λ : Λ πi(Kpn) = 0 for i > pn. 1Kn→Kpn since Kn is obtained from Λ Γ Γ The map λ gives the desired element λ(ι) ∈ H pn( to each fiber K∧p up to homotopy since the restriction map H pn( pn... |
j is a generator of H j(L∞) and θi(α) ∈ H n+i(X). Thus θi increases dimension by i. When p = 2 there is no ambiguity about ωj. For odd p we choose ω1 to be the class dual to the 1 cell of L∞ in its standard cell structure, then we take ω2 to j j 2 and ω2j+1 = ω1ω be the Bockstein βω1 and we set ω2j = ω 2. X It is clear... |
p−1)mn/2λ(α) ` λ(β) for m = |α| and n = |β|. This implies (3) when p = 2 Steenrod Squares and Powers Section 4.L 505 since if we let ω = ω1, hence ωj = ωj, then Sqi(α ` β) ⊗ ωn+m−i = ∇∗ = ∇∗ i X λ(α ` β) = ∇∗ λ(α) ` λ(β) λ(α) ` ∇∗ λ(β) X To show that λ(α ` β) = (−1)p(p−1)mn/2λ(α) ` λ(β) we use the following diagram: = ... |
Kn)∧p includes the (pm + pn) skeleton of (Km ∧ Kn), restriction to this fiber is injective on H pm+pn. On this fiber the two routes around the triangle give (ιm ⊗ ιn)⊗p and ι⊗p n. These differ by a permutation that is the product of (p − 1) + (p − 2) + ··· + 1 = p(p − 1)/2 transpositions of adjacent factors. Since ιm and... |
Homotopy Theory For a generator α ∈ H 1(S 1), consider the element ∇∗(λ(α)) in H 2(RP∞ × S 1) ≈ Hom(H2(RP∞ × S 1; Z2), Z2). A basis for H2(RP∞ × S 1; Z2) is represented by RP2 × {x0} and RP1 × S 1. A cocycle representing ∇∗(λ(α)) takes the value 0 on RP2 × {x0} since RP∞ × {x0} collapses to a point in S 1). On RP1 × S... |
q0 is the identity on the fundamental class ιn, hence Sq0 is the identity on all positive-dimensional classes. Property (7) is proved similarly: Sq1 coincides with the Bockstein β on the generator ω ∈ H 1(RP2) since both equal ω2. Hence Sq1 = β on the iterated suspensions of ω, and the n fold suspension of RP2 is the (... |
Zp act on the second. Factoring out the diagonal action of Zp × Zp on S ∞ × S ∞ × X ∧p2 2X. This projects to L∞× L∞ with a section, and collapsing the section gives 2X. The fibers since the action of Zp × Zp on S ∞× S ∞ is of the projection 2X→L∞× L∞ are X ∧p2 gives a quotient space Γ Λ Λ Λ Steenrod Squares and Powers ... |
Zp × Zp. This is the subgroup of Zp ≀ Zp obtained by restricting the action of the quoΛ tient Zp on Zp p, where this action becomes trivial so that one has the direct product Zp × Zp. p to the diagonal subgroup Zp ⊂ Zp Λ Λ Λ ( 2Kn→Kp2n restricting to ι⊗p2 Λ in a fiber, then extends this over the part of Since it suffices... |
This permutation is a product of p(p − 1)/2 transpositions, one for each pair is sent to (−1)p(p−1)n/2ι⊗p2 (i, j) with 1 ≤ i < j ≤ p, so in a fiber the class ι⊗p2. 2 (ι) = (−1)p(p−1)n/2λ∗ By the uniqueness property of λ2 this means that τ ∗λ∗ 2 (ι). Commutativity of the square then gives P n (−1)p(p−1)n/2∇∗ 2 λ∗ 2 (ι) ... |
+k ⊗ Sqi−kSqj (ι). To write this summation more symmetri- cally with respect to the two ω terms, let n − j + k = 2n − ℓ. Then we get P n−j k i,j,k P n−j n+j−ℓ ω2n−i ⊗ ω2n−ℓ ⊗ Sqi+ℓ−n−jSqj(ι) i,j,ℓ X In view of the symmetry property of ϕr s, which becomes ϕr s = ϕsr for p = 2, switching i and ℓ in this formula leaves it... |
y. = s−j−1 i−2j The final step is to show that 2r +s−j−1 if i < 2s. Both of these i−2j binomial coefficients are zero if i < 2j. If i ≥ 2j then we have 2j ≤ i < 2s, so j < s, hence s − j − 1 ≥ 0. The term 2r then makes no difference in if r is large since this 2r contributes only a single 1 to the dyadic expansion of 2r + ... |
�(s, Xj) = (sr, Xr j) where subscripts are taken mod p and sr means raise each coordinate of s, regarded as a unit vector in C∞, to the r th power and renormalize the resulting vector to have unit length. Then if γ is a generator of the Zp action on S ∞ × X ∧p, we have ϕ(γ(s, Xj)) = ϕ(e2π i/ps, Xj−1) = (e2r π i/psr, Xr... |
�∗(ωℓ) = ωℓ only when the total number of ω1 and ω2 factors in ωℓ is a multiple of p − 1. For ωℓ = ωk 2 this means ℓ = (p − 1)n − i = 2k(p − 1), while for ωℓ = ω1ωk−1 it means ℓ is 1 less than this, 2k(p − 1) − 1. Solving these equations for i gives i = (n − 2k)(p − 1) or i = (n − 2k)(p − 1) + 1. Since n is even this s... |
)p(p−1)mn/2λ(α)λ(β) for |α| = m and |β| = n. From the definition of the θi ’s it then follows that θ0(α ` β) = (−1)p(p−1)mn/2θ0(α)θ0(β), which gives the first part of the claim. The second part follows from this by induction on n. Lemma 4L.15. a1 = ±m! for m = (p − 1)/2, so p = 2m + 1. Proof: It suffices to compute θ0(α) w... |
S 1, since the base space Lp is compact. the complement of the section at infinity in The complement of the section at infinity is a bundle E→Lp with fibers Rp. In general, the one-point compactification of a fiber bundle E over a compact base space with fibers Rn is called the Thom space T (E) of the bundle, and a class in ... |
− 1) since T permutes coordinates cyclically; this amounts to identifying the standard basis vectors v1, ···, vp in Rp with 1, t, ···, tp−1. The polynomial tp − 1 factors over C into the linear factors t − e2π ij/p for j = 0, ···, p − 1. Combining complex conjugate 1≤j≤m (t2 − 2(cos ϕj)t + 1), factors, this gives a fa... |
for E0 which is the product Lp × R, so the projection E0→R one-point compactifies to a map T (E0)→S 1 and we can pull back the chosen generator α ∈ H 1(S 1) to a Thom class for E0. The other Ej ’s have 2 dimensional fibers, which we now view as C rather than R2. Just as Ej is the quotient of S p × C via the identificatio... |
S 1. It is e 512 Chapter 4 Homotopy Theory evident that g is a homeomorphism onto the complement of the point [0, ···, 0, 1] in CPm+1, so sending the point at infinity in T (E1) to [0, ···, 0, 1] gives an extension of g to a homeomorphism T (E1) ≈ CPm+1. Under this homeomorphism the one-point compactifications of the fib... |
back π ∗ j (τj ) in H ∗(E, E − Lp) is a Thom class for E, as one sees by applying the calculation at the end H ∗(T (E)), the of Example 3.11 in each fiber. Under the isomorphism H ∗(E, E − Lp) ≈ j (τj ) corresponds to ±λ(α) since both classes restrict to ±α⊗p in each class fiber S p ⊂ T (E) and λ(α) is uniquely determine... |
degree j on CP1 ⊂ CPm. And q∗(γ) = ±ω2 since q restricts to a homeomorphism on the 2 cell of Lp in the CW structure defined in Example 2.43. Thus ej = ±jω2, and so ` ωp−1. Since τ0 was the pullback of α τ0 via the projection T (E0)→S 1, when we pull τ0 back to Lp × S 1 via ∇ we get 1 ⊗ α, ⊔⊓ so τ0 ` ··· ` em pulls back... |
−1)(p−1)/2 ≡ −1 and hence (p − 1)/2 Using the formulas an = (−1)p(p−1)n(n−1)/4an 1 and a1 = ± (p − 1)/2! we then have a2i = (−1)p(p−1)2i(2i−1)/4 (p − 1)/2!2i = (−1)p[(p−1)/2]i(2i−1)(−1)i(p+1)/2 = (−1)i(p−1)/2(−1)i(p+1)/2 = (−1)ip = (−1)i since p is odd. since p and 2i − 1 are odd Theorem 4L.16. The operations P i satis... |
(i−j)(p−1)(α)θ2j(p−1)(β) j (−1)i−j a−1 m θ2(i−j)(p−1)(α)(−1)j a−1 n θ2j(p−1)(β) X P i−j(α)P j(β) = = X j j X Property (4), the invariance of P i under suspension, follows from the Cartan formula just as in the case p = 2, using the fact that P 0 is the only P i that can be nonzero on 1 dimensional classes, by (5). The ... |
�(L∞) ⊗ H ∗(Kn). So we get βθ2k(ι) = −θ2k+1(ι) and hence βθ2k(α) = −θ2k+1(α) for all α. Note that βθ2k+1 = 0 from the coefficient of ω(p−1)n−2k−1. This also follows from the formula βθ2k = −θ2k+1 since β2 = 0. P P In order to show that β∇∗(λ(ι)) = 0 we first compute βλ(ι). We may assume Kn has a single n cell and a single... |
ϕ⊗i ⊗ ψ ⊗ ϕ⊗(p−i−1) represents βλ(ι). Via the so the formula (∗) says that P Steenrod Squares and Powers Section 4.L 515 ∆ Γ Λ quotient map Kn→ Kn, the class λ(ι) pulls back to a class γ(ι) with βγ(ι) also repi(−1)inϕ⊗i ⊗ ψ ⊗ ϕ⊗(p−i−1). To see what happens when we pull βγ(ι) Γ Kn, consider the commutative diP resented ... |
1 ’ is the cellular cocycle assigning the value 1 to each 0 cell of S ∞. By the definition of τ we have τ(1 ⊗ ψ ⊗ ϕ⊗(p−1)) = n permutes the factors cyclically. It does not matter whether T moves coordinates one unit left- i T i(ψ ⊗ ϕ⊗(p−1)) where T : K∧p n →K∧p P wards or one unit rightwards since we are summing over a... |
∞ is contractible, so ∇∗(βλ(ι)) is in the image of the composition τπ ∗ across the bottom of the diagram. But this composition is multiplication by p, which is zero for Zp coefficients, so β∇∗(λ(ι)) = ∇∗(βλ(ι)) = 0. ⊔⊓ The derivation of the Adem relations now follows the pattern for the case p = 2. We had the formula ∇∗... |
be fixed throughout the discussion, we may factor out the nonzero constant a2mnan. Then applying the Cartan formula to expand the P i terms, using also the formulas P k(ω2r ) = ω2r +2k(p−1)+1 de- rived earlier in the section, we obtain ω2r +2k(p−1) and P k(ω2r +1) = r k r k i,j,k X i,j,k X i,j,k X i,j,k X i,j,k − − + −... |
�) ⊗ P i+ℓ−mn−jP j(ι) and similarly for the other four summations. X 2 λ∗ Now we bring in the symmetry property ϕr s = (−1)r s+mnp ϕsr, where, as before, ∇∗ 2 (ι) = r,s ωr ⊗ ωs ⊗ ϕr s. Of the five summations, only the first has both ω terms with even subscripts, namely r = 2m(pn − 2i) and s = 2m(pn − 2ℓ), so the coefficien... |
so that the left side of (1) has only one term, namely we take n = 2(1 + p + ··· + pr −1) + 2s and ℓ = mn + s for given integers r and s. Then m(n−2j) mn+j−ℓ = pr −1−(p−1)(j−s) j−s and if r is large, this binomial coefficient is 1 if j = s and 0 otherwise since if the rightmost nonzero digit in the p adic expansion of t... |
side of (2) then reduces to (−1)i+s P iβP s(ι) and (2) becomes P iβP s(ι) = j X (−1)i+j − (p−1)(pr +s−j) i−pj βP i+s−jP j(ι) (p−1)(pr +s−j)−1 i−pj−1 (−1)i+j P i+s−jβP j(ι) j X This time the term pr can be omitted if r is large and i ≤ ps. ⊔⊓ Exercises 1. Determine all cohomology operations H 1(X; Z)→H n(X; Z), H 2(X; ... |
a CW complex is a space X constructed in the following way: (1) Start with a discrete set X 0, the 0 cells of X. (2) Inductively, form the n skeleton X n from X n−1 by attaching n cells en ϕα : S n−1→X n−1. This means that X n is the quotient space of X n−1 the identifications x ∼ ϕα(x) for x ∈ ∂Dn image of Dn α under ... |
the definition of the quotient topology on X n. Hence by (3), A is open in X. α for each −1 α (A) is open in Dn Φ −1 α (A) is open in Dn α for each characteristic map Φ Φ Φ A consequence of this characterization of the topology on X is that X is a quotient Φ space of n,α Dn α. ` 520 Appendix Topology of Cell Complexes ... |
subset of the compact set C. Therefore S −1 α (S) consists of at most one more point in Dn α, and Φ must be finite, a contradiction. Since C is contained in a finite union of cells, it suffices to show that a finite union of cells is contained in a finite subcomplex of X. A finite union of finite subcomplexes is again a finite... |
α) is contained in the union of a finite number of cells of dimension less than n. Φ (iii) A subset of X is closed iff it meets the closure of each cell of X in a closed set. Condition (iii) can be restated as saying that a set C ⊂ X is closed iff −1 α (C) is closed in Dn α for all α, since a map from a compact space ont... |
hence C ∩ e m α, so f −1(C) closed implies f −1(C) ∩ Dn α is closed, hence compact, hence its image C ∩ e n α under f is compact and therefore closed. Finally there is the case m > n. Then C ⊂ X n implies C ∩ e m β ). The latter space is contained in a finite union of e ℓ γ ’s with ℓ < m. By induction on m, each C ∩ e ... |
‘spherical’ coordinates (r, θ) in Dn+1, where r ∈ [0, 1] is the radial coordinate and θ lies in ∂Dn+1 = S n. Then we define Nε(A) = ε (A). This is an open set in X since it pulls back to an open set under each characteristic map. α : Dn+1→X of each cell en+1 ε (A), namely, N n+1 ε n N n N n (A) −1 α Φ Φ Φ ε S Propositi... |
contractible. apart. Also, −1 α Φ will make −1 α N n N n and −1 α −1 α −1 α Φ Φ Φ Φ Φ (x) by sliding outward along radial segments in cells en Proof: Given a point x in a CW complex X and a neighborhood U of x in X, we can choose the εα ’s small enough so that Nε(x) ⊂ U by requiring that the closure of N n ε (x) be co... |
B as well as into open sets A and B. α of all cells en A map f : X→Y with domain a CW complex is continuous iff its restrictions to the closures e n α are continuous, and it is useful to know that the same is true for homotopies ft : X→Y. With this objective in mind, let us introduce a little terminology. A topological... |
natural CW structure, but its topology is in general finer, with more open sets, than the product topology. However, the distinctions between the two topologies are rather small, and indeed nonexistent in most cases of interest, so there is no real problem for algebraic topology. Given a space X and a collection of sub... |
the theorem is a special case of Proposition A.15, hav- β. Hence the products ing nothing to do with CW complexes, which says that a product X × Y is compactly generated if X is compactly generated Hausdorff and Y is locally compact. For the last statement of the theorem, suppose X and Y each have at most countably man... |
S We will describe now an example from [Dowker 1952] where the product topology on X × Y differs from the CW topology. Both X and Y will be graphs consisting of Topology of Cell Complexes Appendix 525 infinitely many edges emanating from a single vertex, with uncountably many edges for X and countably many for Y. W Let ... |
this condition occurs in the Lefschetz fixed point theorem, and it was used in the proof of Alexander duality. So let us study this situation in more detail. Theorem A.7. A compact subspace K of Rn is a retract of some neighborhood iff K is locally contractible in the weak sense that for each x ∈ K and each neighborhood... |
implies that the open faces of cubes of A that are minimal with respect to inclusion among such faces form the cells of a CW structure on X, since the boundary of such a face is a union of such faces. The vertices of this CW structure are thus the vertices of all the cubes of A, and the n cells are the interiors of th... |
of B and the definition of r on 0 cells we have r (Y 0) ⊂ V0. Since V0 is contractible in U0, r is defined on the 1 cells of Y. Also, r (Y 1) ⊂ V1 by the definition of r on 1 cells and the fact that U0 is much smaller than V1. Similarly, by induction we have r defined on Y i with r (Y i) ⊂ Vi for all i. In particular, r m... |
K with n vertices is a subcomplex of a simplex n−1, and hence embeds in Rn. The preceding theorem then implies that K is a retract of some neighborhood in Rn, so any retract of K is also a retract of such a ∆ neighborhood, via the composition of the two retractions. Conversely, let K be a compact space that is a retra... |
suffices to show that a finite CW complex can be embedded in some Rn. This is proved by induction on the number 528 Appendix Topology of Cell Complexes of cells. Suppose the CW complex X is obtained from a subcomplex A by attaching a cell ek via a map f : S k−1→A, and suppose that we have an embedding A ֓ Rm. Then we can... |
f4f3, ···). i ` ` by attaching Xi × {i} The second of these is obvious. To prove the other two we will use Proposition 0.18, whose proof applies not just to CW pairs but to any pair (X1, A) for which there is a deformation retraction of X1 × I onto X1 × {0} ∪ A× I. To prove (1) we regard T (f1, f2, ···) as being obtai... |
] complex X via maps Y i-----→ X i ` The Compact-Open Topology Appendix 529 One might ask whether a space dominated by a finite CW complex is homotopy equivalent to a finite CW complex. In the simply-connected case this follows from Proposition 4C.1 since such a space has finitely generated homology groups. But there are ... |
X Y of maps f : Y →X has a subbasis consisting of the sets M(K, U) of mappings taking a compact set K ⊂ Y to an open set U ⊂ X. Thus a basis for X Y consists of sets of maps taking a finite number of compact sets Ki ⊂ Y to open sets Ui ⊂ X. If Y is compact, which is the only case we consider in this book, convergence t... |
) i Ki, and f (Ki) ⊂ Bε/2 i M(Ki, Ui) ⊂ Bε(f ), suppose that g ∈ f (y), f (yi) f (y), f (yi) g(y), f (yi) g(y), f (yi) Bε/3(f (yi)) = Ui, hence f ∈ f (y), g(y) ≤ d + d T T T S Conversely, we show that for each open set M(K, U) and each f ∈ M(K, U) there is a ball Bε(f ) ⊂ M(K, U). Since f (K) is compact, it has a dista... |
epoint-preserving map Σ a map f : I × X→Y taking ∂I × X ∪ {x0}× I to the basepoint of Y, so the associated X→Y gives map a map F : I × X × I→Y taking ∂I × X × I ∪ I × {x0}× I to the basepoint, with F a map b X × I→ Y ⊂ Y I defining a basepoint-preserving homotopy f : X→Y I has image in the subspace Y ⊂ Y I. A homotopy f... |
��rst: Proposition A.15. If X is a compactly generated Hausdorff space and Y is locally compact, then the product topology on X × Y is compactly generated. Proof: First a preliminary observation: A function f : X × Y →Z is continuous iff its restrictions f : C × Y →Z are continuous for all compact C ⊂ X. For, using (b) a... |
and B range over compact sets in Y and Z respectively and U ranges over open sets in X. Given a compact K ⊂ Y × Z and a map f ∈ M(K, U), let KY and KZ be the projections of K onto Y and Z. Then KY × KZ is compact Hausdorff and 532 Appendix The Compact-Open Topology hence normal. A normal space has the property that for... |
. Then we let Q = X Y with subbasis the sets M(A, U). Given f ∈ M(K, U) with K compact in Z and U open in Q, write U as a union of basic sets Uα with each Uα an intersection of finitely many sets Vα,j of the given subbasis for Q. The cover of K by the open sets f −1(Uα) has a finite subcover, say by the open sets f −1(Ui... |
Appendix 533 Since g is continuous, so is the associated map g : X→W Z, by Proposition A.14. h : Y →W Z is continuous since f is a quotient map. Applying Propo⊔⊓ b sition A.14 again, we conclude that h is continuous. This implies that b The Homotopy Extension Property Near the end of Chapter 0 we stated, and partially... |
set Un follows from the fact that a union of open sets with this property is again an open set with this property. Let U = n Un. Note that A∩O ⊂ U since O intersects A× I in an open set in A× I by assumption. It will suffice to show that x ∈ U since if x ∈ Un then we can choose V × W = (Un ∩ O)× [0, 1/n) ∩ Y ⊂ O, where ... |
, so by letting t approach 0 we conclude that r1(x, 0) ∈ A − O since r1 is a continuous map to X and X ∩ O is open in X. Since r1(x, 0) = x we deduce that x is not in O. However, this contradicts the fact that x was chosen to be a point in O. From this contradiction we conclude that x must be in U, and the proof is fini... |
no longer pay attention to orderings of vertices of simplices, we obtain a weaker structure which could be called an unordered D complex. Here each cell en α n→X, but the restriction of σα to a face has a distinguished characteristic map σα : n−1→X with a symmetry of of permuting its vertices. Alternatively, we could ... |
∆ complex in which each simplex is uniquely ∆ ∆ determined by its vertices. In the literature a regular unordered complex is some- times called a simplicial multicomplex, or just a multicomplex, to convey the idea that there can be many simplices with the same set of vertices. The barycentric subdivi- sion of a regula... |
ices in each face ∆ ∆ ∆ 536 Appendix Simplicial CW Structures n. Conversely, any CW complex built in this way is an s of the usual CW structure on S n consisting of one 0 cell and one n cell is an s structure since the attaching map of the n cell, the constant map, is a simplicial complex. For example, complex ∆ ∆ n to... |
g∗. Such a functor is exactly equivalent to a Explicitly, we can reconstruct the n complex. ∆ ∆ complex X from the functor by setting ∆ X = ∆ n(Xn × n)/(g∗(x), y) ∼ (x, g∗(y)) ` for (x, y) ∈ Xn × of k to the g(i)th vertex of ∆ k, where g∗ is the linear inclusion k→ n sending the ith vertex n, and we perform the indica... |
in fact it has the additional structure of a simplicial set. ∆ ∆ In a similar but more restricted way, an s (X) whose k simplices are all the simplicial maps complex X gives rise to a simplicial k→X. These are uniquely set n→X of simplicial surjections q (preservexpressible as compositions σαq : ing orderings of verti... |
degenerate simplex of Y has the form g∗(τ) for some nondegenerate simk. We claim that such a g and τ are unique. For n→ 1 (τ1) = g∗ 2 (τ2) with τ1 and τ2 nondegenerate and g1 : surjective. Choose order-preserving injections h1 : ∆ plex τ and surjection g : n→ suppose we have g∗ k1 k1→ and g2 : n and ∆ ∆ k2→ h2 : 2 (τ2... |
injection. Letting g range over surjections amounts to ` k and g : k→ ∆ ∆ ∆ collapsing each simplex onto a unique nondegenerate simplex by a unique projection, by the claim in the preceding paragraph, so after performing the identifications just for surjections we obtain a collection of disjoint simplices, with one n s... |
this fact here. The |Y | is sometimes called the thick geometric realization of Y. complex |Y ∆ ∆ ∆ ∆ Since simplicial sets are very combinatorial objects, many standard constructions ∆ can be performed on them. A good example is products. For simplicial sets X and Y there is an easily-defined product simplicial set X ... |
where one factor is a degenerate 1 simplex and the other is a nondegenerate 1 simplex. Obviously there are no nondegenerate n simplices in X × Y for n > 2. ∆ It is not hard to see how this example generalizes to the product q. Here one obtains the subdivision of the product into (p + q) simplices described in §3.B p ×... |
Algebraic Topology, Springer-Verlag GTM 82, 1982. G. E. Bredon, Topology and Geometry, Springer-Verlag GTM 139, 1993. K. S. Brown, Cohomology of Groups, Springer-Verlag GTM 87, 1982. R. F. Brown, The Lefschetz Fixed Point Theorem, Scott Foresman, 1971. M. M. Cohen, A Course in Simple-Homotopy Theory, Springer-Verlag G... |
Introduction to Homotopy Theory, Cambridge Univ. Press, 1953. P. J. Hilton, Homotopy Theory and Duality, Gordon and Breach, 1966. P. J. Hilton and S. Wylie, Homology Theory, Cambridge Univ. Press, 1967. P. J. Hilton and U. Stammbach, A Course in Homological Algebra, Springer-Verlag GTM 4, 1970. P. J. Hilton, G. Mislin... |
ˇsek, Using the Borsuk-Ulam Theorem, Springer-Verlag, 2003. 542 Bibliography C. R. F. Maunder, Algebraic Topology, Cambridge Univ. Press, 1980 (reprinted by Dover Pub- lications). J. P. May, Simplicial Objects in Algebraic Topology, Van Nostrand, 1967 (reprinted by Univ. Chicago Press). J. P. May, A Concise Course in A... |
Princeton Univ. Press, 1968. D. Sullivan, Geometric Topology, xeroxed notes from MIT, 1970. R. M. Switzer, Algebraic Topology, Springer-Verlag, 1975. H. Toda, Composition Methods in Homotopy Groups of Spheres, Ann. of Math. Studies 49, 1962. K. Varadarajan, The Finiteness Obstruction of C. T. C. Wall, Wiley, 1989. C. ... |
fibr´es principaux, Ann. of Math. 57 (1953), 115–207. R. Bott and H. Samelson, On the Pontryagin product in spaces of paths, Comment. Math. Helv. 27 (1953), 320–337. R. Bott and J. Milnor, On the parallelizability of spheres, Bull. A.M.S. 64 (1958), 87–89. R. Bott, The stable homotopy of the classical groups, Ann. of M... |
aspherical spaces, J. Austral. Math. Soc. 16 (1973), 332–352. 544 Bibliography R. D. Edwards, A contractible, nowhere locally connected compactum, Abstracts A.M.S. 20 (1999), 494. S. Eilenberg, Singular homology theory, Ann. of Math. 45 (1944), 407–447. S. Eilenberg and J. A. Zilber, Semi-simplicial complexes and sing... |
homology and the Triangulation Con- jecture, Jour. A.M.S. 29 (2016), 147–176. J. P. May, A general approach to Steenrod operations, Springer Lecture Notes 168 (1970), 153– 231. J. P. May, Weak equivalences and quasifibrations, Springer Lecture Notes 1425 (1990), 91–101. C. Miller, The topology of rotation groups, Ann. ... |
��brations, Ill. J. Math. 16 (1972), 234–242. P. Scott and T. Wall, Topological methods in group theory, London Math. Soc. Lecture Notes 36 (1979), 137–203. J.-P. Serre, Homologie singuli`ere des espaces fibr´es, Ann. of Math. 54 (1951), 425–505. J.-P. Serre, Groupes d’homotopie et classes de groupes ab´eliens, Ann. of ... |
technical restrictions on what subsets can or cannot be events, according to the mathematical subject of measure theory. But we will not concern ourselves with such technicalities here.) Finally, and most importantly, a probability model requires a probability measure, usually written P. This probability measure must ... |
.2: Probability Models EXAMPLE 1.2.1 Consider again the weather example, with S rain, snow, clear. Suppose that the probability of rain is 40%, the probability of snow is 15%, and the probability of a clear day is 45%. We can express this as P rain 0 15, and P clear 0 40, P snow 0 45. 0, i.e., it is impossible that not... |
more complicated (e.g., uncountably infinite) sample spaces as well. EXAMPLE 1.2.6 Suppose that S S by saying that [0 1] is the unit interval. We can define a probability measure P on P [a b] b a whenever 0 a b 1 (1.2.2) Chapter 1: Probability Models 7 In words, for any1 subinterval [a b] of [0 1], the probability of th... |
orous Probability Theory, Second Edition, by J. S. Rosenthal (World Scientific Publishing, Singapore, 2006). 8 The union Section 1.2: Probability Models A B s : s A or s B of the sets A and B is the set of elements that are in either A or B In Figure 1.2.2, it is depicted by the region covered by both circles. Notice th... |
events A such that P A 1.2.2 Suppose S (a) What is P 1 2? (b) What is P 1 2 3? (c) How many events A are there such that P A 1.2.3 Suppose S P 2 be? 1.2.4 Suppose S 0 7, P 1 3 0 5, P 2 3 P a valid probability measure? Why or why not? 1.2.5 Consider the uniform distribution on [0 1]. Let s is P s? Do you find this resul... |
1.2.15 Suppose S is an uncountable set. Is it possible that P s s S? Why or why not? S? Why or why not? 0 for every 0 for every single DISCUSSION TOPICS 1.2.16 Does the additivity property make sense intuitively? Why or why not? 1.2.17 Is it important that we always have P S change if this were not the case? 1? How wo... |
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