Contents 41 Suspensions and shift 1 42 The telescope construction 10 43 Fibrations and cofibrations 15 44 Cofibrant generation 22 41 Suspensions and shift The suspension X ^S1 and the fake suspension SX of a spectrum X were defined in Section 35 — the constructions differ by a non-trivial twist of bond- ing maps. The loop spectrum for X is the function complex object 1 hom∗(S ;X): There is a natural bijection 1 ∼ 1 hom(X ^ S ;Y) = hom(X;hom∗(S ;Y)) so that the suspension and loop functors are ad- joint. The fake loop spectrum WY for a spectrum Y con- sists of the pointed spaces WY n, n ≥ 0, with adjoint bonding maps n 2 n+1 Ws∗ : WY ! W Y : 1 There is a natural bijection hom(SX;Y) =∼ hom(X;WY); so the fake suspension functor is left adjoint to fake loops. n n+1 The adjoint bonding maps s∗ : Y ! WY define a natural map g : Y ! WY[1]: for spectra Y. The map w : Y ! W¥Y of the last section is the filtered colimit of the maps g Wg[1] W2g[2] Y −! WY[1] −−−! W2Y[2] −−−! ::: Recall the statement of the Freudenthal suspen- sion theorem (Theorem 34.2): Theorem 41.1. Suppose that a pointed space X is n-connected, where n ≥ 0. Then the homotopy fibre F of the canonical map h : X ! W(X ^ S1) is 2n-connected. 2 In particular, the suspension homomorphism 1 ∼ 1 piX ! pi(W(X ^ S )) = pi+1(X ^ S ) is an isomorphism for i ≤ 2n and is an epimor- phism for i = 2n+1, provided that X is connected. In general (ie. with no connectivity assumptions on Y), the space Sn ^Y is (n − 1)-connected, by Lemma 31.5 and Corollary 34.1. Thus, the suspension homomorphism n+k n+k+1 pi+k(S ^Y) ! pi+k+1(S ^Y) is an isomorphism if i ≤ 2n − 2 + k, and it follows that the map n s ¥ pi(S ^Y) ! pi−n(S Y) is an isomorphism for i ≤ 2(n − 1). Here’s an easy observation: Lemma 41.2. The natural map g : X ! WX[1] is a stable equivalence if X is strictly fibrant. Proof. This is a cofinality argument, which uses the fact that W¥X is the filtered colimit of the sys- tem X ! WX[1] ! W2X[2] ! ::: 3 Lemma 41.3. Suppose that Y is a pointed space. Then the canonical map h : S¥Y ! WS(S¥Y) is a stable homotopy equivalence. Proof. The map n s ¥ pk(S ^Y) ! pk−n(S Y) is an isomorphism for k ≤ 2(n − 1). Similarly (exercise), the map n+1 s ¥ pk(W(S ^ X)) ! pk−n(WS(S X)) is an isomorphism for k + 1 ≤ 2n or k ≤ 2n − 1. There is a commutative diagram n =∼ s ¥ pk(S ^Y) / pk−n(S Y) =∼ n+ s 1 / ¥ pk(W(S ^Y)) =∼ pk−n(WS(S Y)) in which the indicated maps are isomorphisms for k ≤ 2(n − 1). It follows that the map s ¥ s ¥ pp(S Y) ! pp(WS(S Y)) is an isomorphism for p ≤ n − 2. Finish by letting n vary. 4 Remark: What we’ve really shown in Lemma 41.3 is that the composite h W j S¥X −! WS(S¥X) −! WF(S(S¥X)) is a natural stable equivalence. Lemma 41.4. Suppose that Y is a spectrum. Then the composite h W j Y −! WSY −! WF(SY) is a stable equivalence. Proof. We show that the maps h W j LnY −! WSLnY −! WF(SLnY) arising from the layer filtration for Y are stable equivalences. In the layer filtration 0 n 1 n 2 n LnY : Y ;:::;Y ;S ^Y ;S ^Y ;::: the maps ¥ n r r (S Y [−n]) ! LnY are isomorphisms for r ≥ n. Thus, the maps ¥ n r r (WF(S(S Y [−n]))) ! WF(S(LnY)) 5 are weak equivalences for r ≥ n, so that ¥ n WF(S(S Y [−n])) ! WF(S(LnY)) is a stable equivalence. The map h : X ! WSX respects shifts, so Lemma 41.3 implies that the composite S¥Y n[−n] ! WS(S¥Y n[−n]) ! WF(S(S¥Y n[−n])) is a stable equivalence. Theorem 41.5. Suppose that X is a spectrum. Then the canonical map s : SX ! X[1] is a stable equivalence. Proof. The map s is adjoint to the map s∗ : X ! WX[1], so that there is a commutative diagram h W j X / WSX / WF(SX) s Ws WFs ∗ " X[ ] / F(X[ ]) W 1 W j W 1 where j : SX ! F(SX) is a strictly fibrant model. The composite W j[1] X −!s∗ WX[1] −−−! W(FX)[1] 6 is a stable equivalence by Lemma 41.2, and the shifted map j[1] : X[1] ! (FX)[1] is a strictly fi- brant model of X[1]. It follows that the composite W j X −!s∗ WX[1] −! WF(X[1]) is a stable equivalence. The composite h W j X −! WSX −! WF(SX) is a stable equivalence by Lemma 41.4. The map WFs is therefore a stable equivalence, so Lemma 41.2 implies that Fs : F(SX) ! F(X[1]) is a stable equivalence. Here’s another, still elementary but much fussier result: Theorem 41.6. The functors X 7! X ^S1 and X 7! SX are naturally stably equivalent. Sketch Proof: ([2], Lemma 1.9, p.7) The isomor- phisms t : S1 ^X n ! X n ^S1 and the bonding maps s ^ S1 together define a spectrum with the space 7 S1 ^ X n in level n, and with bonding maps s˜ de- fined by the diagrams 1 1 n s˜ 1 n+1 S ^ S ^ X / S ^ X S1^t =∼ =∼ t 1 n 1 n+1 1 S ^ X ^ S / X ^ S s^S1 There are commutative diagrams S1 ^ S1 ^ X n S1^s + 1 n+1 t^Xn S ^ X 3 S1 ^ S1 ^ X n s˜ Composing then gives a diagram S1 ^ S1 ^ S1 ^ X n (S1^s)(S1^S1^s) * (3;2;1)^Xn S1 ^ X n+2 4 s˜ ·(S1^s˜ ) S1 ^ S1 ^ S1 ^ X n where (3;2;1) is induced on the smash factors mak- ing up S3 by the corresponding cyclic permutation of order 3. The spaces S1 ^ X 0;S1 ^ X 2;::: and the respective composite bonding maps (S1 ^s)(S1 ^S1 ^s) and s˜ · (S1 ^ s˜ ) define “partial” spectrum structures from which the stable homotopy types of the orig- inal spectra can be recovered. 8 The self map (3;2;1) of the 3-sphere S3 has degree 1 and is therefore homotopic to the identity. This homotopy can be used to describe a telescope construction (see [2], p.11-15, and the next sec- tion) which is stably equivalent to both of these partial spectra. Remark: The proof of Theorem 41.6 that is sketched here is essentially classical. See Prop. 10.53 of [3] for a more modern alternative. Corollary 41.7. 1) The functors X 7! X[1],X 7! SX and X 7! X ^ S1 are naturally stably equiva- lent. 2) The functors X 7! X[−1],X 7! WX and X 7! 1 hom∗(S ;X) are naturally stably equivalent. Proof. Lemma 41.2 implies that the composite W j[1] X −!s∗ WX[1] −−−! WFX[1] is a stable equivalence for all spectra X, where j : X ! FX is a strictly fibrant model. Shift preserves stable equivalences, so the induced composite s [−1] W j X[−1] −−−!∗ WX −! WFX 9 is a stable equivalence. The natural stable equivalence SY 'Y ^S1 induces a natural stable equivalence 1 WX ' hom∗(S ;X) for all strictly fibrant spectra X. In other words, the suspension and loop functors (real or fake) are equivalent to shift functors, and define equivalences Ho(Spt) ! Ho(Spt) of the sta- ble category. 42 The telescope construction Observe that a spectrum Y is cofibrant if and only if all bonding maps s : S1 ^Y n ! Y n+1 are cofi- brations. The telescope TX for a spectrum X is a natural cofibrant replacement, equipped with a natural strict equivalence s : TX ! X. The construction is an iterated mapping cylinder. We find natural trivial cofibrations j a X k −!k CX k −!k TX k; k ≤ n; 10 k k and tk : TX ! X such that tk ·(ak · jk) = 1 and the maps tk define a strict weak equivalence of spectra t : TX ! X. 0 0 0 • X = CX = TX and j0 and a0 are identities, • CX n is the mapping cylinder for s : S1 ^ X n ! X n+1, meaning that there is a pushout diagram 1 n s n+1 S ^ X / X d0 jn+1 1 n 1 n+1 (S ^ X ) ^ D+ / CX zn+1 for each n. Write s∗ for the composite 1 1 n d 1 n 1 zn+1 n+1 S ^ X −! (S ^ X ) ^ D+ −−! CX and observe that s∗ is a cofibration. The projection map 1 n 1 1 n s : (S ^ X ) ^ D+ ! S ^ X 0 satisfies s · d = 1 and induces a map sn+1 : n+1 n+1 CX ! X such that sn+1 · jn+1 = 1. Fur- ther sn+1 · s∗ = s. 11 • Form the pushout diagram 1 n s∗ n+1 S ^ X / CX 1 S ^ jn 1 n S ^CX an+1 1 S ^an 1 n n+1 S ^ TX / TX s˜ Then s˜ is a cofibration, and the maps jn+1, an+1 are trivial cofibrations.