Some variations on the telescope conjecture

Contemporary Mathematics Volme 00, 0000 Some variations on the telescope conjectre DOUGLAS C. RAVENEL November 29, 1993 Abstract. This paper presents some speclations abot alternatives to the recently disproved telescope conjectre in stable homotopy theory. It incldes a brief introdction to the parametrized Adams spectral seqence, the main technical tool sed to disprove it. An example spporting the ne conjectres is described. 1. Introdction A p-local finite spectrm X is said to have type n if K(n 1) (X) =0and K(n) (X) 0. The periodicity theorem of Hopkins-Smith [HS] saysthatany sch complex admits a map Σ d X f X sch that K(n) (f) is an isomorphism and K(m) (X) = 0 for all m>n. Sch a map is called a v n -map. This map is not niqe, bt the direct limit X of the system X f Σ d X f Σ 2d X f, called the telescope associated ith X, is independent of the choice of f. (See [Rav92a] for more backgrond.) This telescope is of interest becase its homotopy grops, nlike those of X itself, are comptable. By this e mean that it is possible in many interesting cases to give a complete explicit description of π ( X). We remind the reader that there is not a single example of a noncontractible finite spectrm X for hich π (X) is completely knon. The only finite complexes X for hich the nstable homotopy grops are completely knon are the ones (sch as srfaces of positive gens) hich happen to be K(π, 1)s. 1991 Mathematics Sbject Classification. Primary 55Q10, 55T14; Secondary 55N20, 55P60. Partially spported by the National Science Fondation and MSRI c 0000 American Mathematical Society 0000-0000/00 $1.00 + $.25 per page 1

2 DOUGLAS C. RAVENEL In the case n = 0, i.e., hen H (X; Q) 0, X is the stable rational homotopy type of X. We have knon ho to compte its homotopy grops for decades. For n =1,π ( X) is called the v 1 -periodic homotopy of X. Since 1982 this has been compted in may interesting cases; these reslts are srveyed by Davis in [Dav]. In this paper e ill speclate here abot this problem for n 2, offering some sbstittes for the original telescope conjectre [Rav84]. The classical Adams spectral seqence is seless for compting π ( X) becase H ( X; Z/(p)) = 0. The Adams-Novikov spectral seqence for π ( X) hasnice properties and can be completely analyzed in many interesting cases, bt is not knon to converge for n>1. The telescope conjectre of 1977 has three eqivalent formlations: The Adams-Novikov spectral seqence for π ( X) converges. The natral map X L n X is an eqivalence, here L n denotes Bosfield localization ith respect to (eqivalently) E(n), vn 1 BP or K(0) K(1) K(n). For X as above, L n X is the same as L K(n) X. X has the same Bosfield class as K(n). These statements are easily proved for n = 0, knon to be tre (bt not easily proved) for n = 1 (this is de to Mahoald [Mah82] forp = 2 and to Miller [Mil81] forp>2), and knon to be false ([Rav92b] and[rava]) for n =2. We ill indicate ho far off the telescope conjectre is for n 2 by describing or best gesses for the vales of π ( X) andπ (L n X) in the simplest cases. What follos is not intended to be a precise statement, bt rather an indication of the flavor of the calclations. They have been verified for n =2andp =2in recent joint ork ith Mahoald and Shick [MRS]. With these caveats in mind, π (L n X) (for a sitable type n finite ring spectrm X) is a sbqotient of an exterior algebra on n 2 generators, hile π ( X) a sbqotient of an exterior algebra on only ( ) ( n+1 2 generators tensored ith n 2) factors of the form Z/(p)[Q/Z (p) ]. (Note that ( ) ( n+1 2 + n ) 2 = n 2.) The appearance of this second type of factor, in place of ( n 2) of the exterior factors in π (L n X), is a startling development. It implies that in π (V (1)) for p 5 (here V (1) stands for Toda s example of a type 2 complex [Tod71]) there is a family of elements x 1,x 2,, each having positive Adams-Novikov filtration, sch that any prodct of them (ith no repeated factors) is nontrivial. We are not aare of any example of this sort that as knon previosly.

VARIATIONS ON THE TELESCOPE CONJECTURE 3 2. The Adams spectral seqence The Adams spectral seqence for π (X) is derived from the folloing Adams diagram. X X 0 X 1 X 2 (1) g 0 g 1 g 2 K 0 K 1 K 2 Here X s+1 is the fibre of g s. We get an exact cople of homotopy grops and a spectral seqence ith E s,t 1 = π t s (K s ) and d r : Er s,t Er s+r,t+r 1. This spectral seqence converges to π (X) if the homotopy inverse limit lim X s is contractible. When X is connective, it is a first qadrant spectral seqence. For more backgrond, see [Rav86]. In the classical Adams spectral seqence e have K s = X s H/p, hereh/p denotes the mod p Eilenberg-Mac Lane spectrm, and in the Adams-Novikov spectral seqence e have K s = X s BP. In each case e have convergence and E 2 can be identified as an Ext grop hich can be compted algebraically. 3. The localized Adams spectral seqence The localized Adams spectral seqence (originally de to Miller [Mil81]) is derived from the Adams spectral seqence in the folloing ay. The telescope X is obtained from X by iterating a v n -map f : X Σ d X. Sppose that this map has positive Adams filtration (hich it alays does in the classical case), ie sppose there is a lifting f : X Σ d X s0 for some s 0 > 0. This ill indce maps f : X s Σ d X s+s0 enablesstodefine X s to be the homotopy direct limit of for s 0. This f f X s Σ d X s+s0 Σ 2d X s+2s0 X s = X for s<0. Ths e get the folloing diagram, generalizing that of (1). X 1 X0 X1 f g 1 g 0 g 1 K 1 K0 K1,

4 DOUGLAS C. RAVENEL here the spectra K s are defined after the fact as the obvios cofibres. This leads to a fll plane spectral seqence (the localized Adams spectral seqence) ith E s,t 1 = π t s ( K s ) and d r : Er s,t Er s+r,t+r 1 as before. This spectral seqence converges to the homotopy of the homotopy direct limit π (lim X s ) if the homotopy inverse limit lim Xs is contractible. The folloing reslt is proved in [Ravb]. Theorem 2(Convergence of the localized Adams spectral seqence). For a type n finite complex X, in the localized Adams spectral seqence for π ( X) e have The homotopy direct limit lim X s is the telescope X. The homotopy inverse limit lim Xs is contractible if the original (nlocalized) Adams spectral seqence has a vanishing line of slope s 0 /d at E r for some finite r, i.e., if there are constants c and r sch that E s,t r =0 for s>c+(t s)(s 0 /d). (In this case e say that f has a parallel lifting f.) The proof of this reslt is not deep; it only involves figring ot hich diagrams to chase. Here are some informative examples. If e start ith the Adams-Novikov spectral seqence, then the map f cannot be lifted since BP (f) is nontrivial. Ths e have s 0 =0and the lifting condition reqires that X has a horizontal vanishing line in its Adams-Novikov spectral seqence. This is not knon (or sspected) to occr for any nontrivial finite X, so e do not get a convergence theorem abot the localized Adams-Novikov spectral seqence, hich is merely the standard Adams-Novikov spectral seqence applied to X. Thislack of content is to be expected given the depth of the reslt. If e start ith the classical Adams spectral seqence, an npblished theorem of Hopkins-Smith says that a type nxalays has a vanishing line of slope 1/ v n =1/(2p n 2). Ths e have convergence if f has a lifting ith s 0 = d/ v n. This does happen in the fe cases here Toda s complex V (n) exists. Then V (n 1) is a type n complex ith a v n -map ith d = v n and s 0 =1. The lifting described above does not exist in general. For example, let X be the mod 4 Moore spectrm. It has type 1 and v 1 =2. Adams constrcted a v 1 -map f ith d = 8, bt its filtration is 3, rather than 4 as reqired by the convergence theorem. Replacing f ithaniterate f i does not help, becase its filtration is only 4i 1. This difficlty can be fixed ith the localized parametrized Adams spectral seqence (see [Ravb] for more details), to be described belo.

VARIATIONS ON THE TELESCOPE CONJECTURE 5 In favorable cases (sch as Toda s examples) the E 2 -term of the localized Adams spectral seqence can be identified as an Ext grops hich can be compted explicitly. 4. The parametrized Adams spectral seqence We ill describe a family of Adams spectral seqences parametrized by a rational nmber ɛ interpolating beteen the classical Adams spectral seqence (the case ɛ = 1) and the Adams-Novikov spectral seqence (ɛ = 0). The constrction is easy to describe, bt difficlt to carry ot in detail. First e need some notation. Let G = BP and F = H/p. We have the folloing homotopy commtative diagram in hich the ros are cofibre seqences, and h and h are the sal nit maps. r G S 0 G h ι ι r h F S 0 F Then in the Adams diagram for the classical Adams spectral seqence e have X s = X F (s), and in the one for the Adams-Novikov spectral seqence, X s = X G (s). We ant to smash these to Adams diagram together and get a 2-dimensional diagram, bt e also ant to exploit the map t above. For i, j 0let X i,j = X { S 0 for j i F (j i) for j>i } G (i) We ill define maps X i,j γ i,j X i 1,j for i>0, and X i,j ϕ i,j X i,j 1 for j>0 by { γ i,j = X { ϕ i,j = X } F (j i) ι if j i G (i 1) r otherise } F (j i 1) r if j>i G (i) S 0 otherise

6 DOUGLAS C. RAVENEL It follos that e have a diagram γ 1,0 γ 2,0 X X 0,0 X 1,0 X 2,0 ϕ 0,1 ϕ 1,1 ϕ 2,1 γ 1,1 γ 2,1 X 0,1 X 1,1 X 2,1 ϕ 0,2 ϕ 1,2 ϕ 2,2 γ 1,2 γ 2,2 X 0,2 X 1,2 X 2,2 This diagram commtes p to homotopy and is eqivalent to one that commtes strictly. Hence it makes sense to speak of nions and intersections of the varios X i,j as sbspectra of X. No fix a nmber 0 ɛ 1, and for each s 0let X s = X i,j = X G (i) F (j). (1 ɛ)i+ɛj s i+ɛj s Definition 3. For a rational nmber ɛ = k/m (ith m and k relatively prime) beteen 0 and 1, the (G, F )-based Adams spectral seqence parametrized by ɛ (or parametrized Adams spectral seqence for short) is the homotopy spectral seqence based on the exact cople associated ith the resoltion X X 0 X 1/m X 2/m g 0 g 1/m g 2/m K 0 K 1/m K 2/m ith X s as above and K s the obvios cofibre. This is a reindexed form of the Adams diagram of 2; the index s need not be an integer bt ill alays be a hole mltiple of 1/m. Ths e have E s,t 1/m = π t s(k s/m ) and d r : Er s,t Er s+r,t+r 1. The indices r, s and t need not be integers here, bt t s (the topological dimension) is alays a hole nmber. In favorable cases, hen the classical Adams spectral seqence for BP (X) = π (BP X) collapses from E 2 (as it does hen X isatodacomplex),ecan describe E 1+ɛ in terms of Ext grops. This parametrized Adams spectral seqence can be localized the same ay the classical Adams spectral seqence can be, and there is a similar convergence

VARIATIONS ON THE TELESCOPE CONJECTURE 7 theorem for the localized parametrized Adams spectral seqence. Ths e need to examine the existence of parallel liftings again. The methods of Hopkins-Smith can be adopted to this sitation to sho that for any ɛ>0, the parametrized Adams spectral seqence for a type n complex X has a vanishing line of slope ɛ/ v n. We also have the folloing reslt. Theorem 4. Let f : X Σ d X be a v n -map. Then for ɛ< v n /d, f has a lifting to Σ d X ɛd/ vn. Hence for any type n complex X, the localized parametrized Adams spectral seqence converges to π ( X) for sfficiently small positive ɛ. 5. Some conjectres No e ill speclate abot the behavior of this localized parametrized Adams spectral seqence converging to π ( X). Each statement e ill make belo has been verified in a special case here n =2andp = 2, in recent joint ork ith Mahoald and Shick [MRS]. Presmably these statements ill be proved by shoing that the indicated properties are generic, i.e., the set of spectra having them is closed nder cofibrations and retracts. Then the thick sbcategory theorem [Rav92a, Chapter 5], hich classifies all sch sets of finite spectra, ill say that if they are tre for one type n complex, they are tre for all of them. Hoever, at the moment e cannot prove that any of the properties e ill discss are generic. We have three conjectres. All concern the behavior of the localized parametrized Adams spectral seqence converging to π ( X) for sfficiently small positive ɛ. Conjectre 5. The localized parametrized Adams spectral seqence collapses from E r for some finite r. No recall that or spectral seqence has a vanishing line of slope ɛ/ v n at E.Letc be its s-intercept, that is the smallest nmber sch that E s,t =0 for s>c+ ɛ(t s)/ v n. Recall also that the v n -map f indces isomorphisms E s,t r E s+ɛ(d/ vn ),t+d+ɛ(d/ vn ) r commting ith differentials, so E is determined by E s,t for 0 t s<d. With this in mind, let ρ(y) denote the total rank of the E s,t ith 0 t s <d and c + ɛ(t s)/ v n s>c y + ɛ(t s)/ v n. In the sal chart (ith horizontal coordinate t s) and vertical coordinate s, this is a parallelogram shaped region bonded by to vertical lines d nits apart, the vanishing line, and a line y nits belo and parallel to it.

8 DOUGLAS C. RAVENEL Conjectre 6. Let ρ(y) be as above. gros assymptotically ith y (n 2). Then it is finite for all y 0 and This groth estimate for ρ(y) is related the ( n 2) factors of the form Z/(p)[Q/Z(p) ]. If the telescope conjectre ere tre, ρ(y) old be bonded. Or third conjectre concerns the behavior of the spectral seqence as ɛ approaches 0. This type of analysis as crcial in the disproof of the telescope conjectre. Given an element x E s,t, e define its effective filtration φ(x, ɛ) by r φ(x, ɛ) =s v n (t s). ɛ On the chart, this is the s-intercept of a line parallel to the vanishing line, throgh the point corresponding to x. It is invariant nder composition ith f. For x π ( X), for each sfficiently small ɛ>0, there is a niqe nontrivial permanent cycle x ɛ represented by x, and e define φ(x, ɛ) tobeφ(x ɛ,ɛ). Conjectre 7. For x π ( X), let φ(x, ɛ) be as above. Then the qantity λ 0 (x) = lim φ(x, ɛ) ɛ 0+ is either + (in hich case e say x is parabolic), or it is an integer ranging from 0 to ( ) n+1 2, in hich case e say that x is linear. A nontrivial element is linear if and only if its image in π (L n X) is nontrivial, and λ 0 (x) is its Adams-Novikov filtration. For a parabolic element x, the fnction ɛφ(x, ɛ) is bonded, and e define λ 1 (x) = lim sp ɛφ(x, ɛ), ɛ 0+ λ 2 (x) = lim sp ɛ 0+ ( φ(x, ɛ) λ ) 1(x), ɛ and µ(x) = lim sp ɛ 0+ ɛφ(x, ɛ) lim inf ɛ 0+ ɛφ(x, ɛ). (We call these qantities the focal length, displacement and magnification of x respectively.) These qantities are all nonnegative and finite, and sbject to bonds depending only on p and n. For n =2, 0 <λ 1 (x) < p2 1 (p+1)2 p, µ(x) = 4p,andλ 2 (x) λ 1 (x)/(p 1) is an integer ranging from 0 to ( ) n+1 2. 6. An example ith n =2 We ill no describe an example that illstrates this conjectre and motivates its terminology. The relevant comptations ere done in [Rava]. Let X = V (1)

VARIATIONS ON THE TELESCOPE CONJECTURE 9 for p 5. It has a v 2 -map f of degree v 2. Let x π ( X) be the image of the composite h S v2 Σ v2 V (1) f j V (1) S 2p Σ 2p V (1) here h and j are the obvios inclsion and pinch maps. In the localized parametrized Adams spectral seqence e have elements b i,0 E 2,2p(pi 1) 1+ɛ and h i,1 E 1,2p(pi 1) 1+ɛ for i>0. For large ɛ, b 1,0 is the nontrivial permanent cycle represented by x, bt for small ɛ, b 1,0 is killed by a differential and the sitation is more complicated. For i>0, let ɛ i = 2p 2 p i 1 = 2 1+p + + p i 1 2 p i 1 Then for i>1 e have differentials v 2 b p i,0 for ɛ>ɛ i d r (h i+1,1 )= v 2 b p i,0 ± vpi 2 b i 1,0 for ɛ = ɛ i v pi 2 b i 1,0 for ɛ<ɛ i for sitable vales of r. The element h 1,1 is alays a permanent cycle represented by a linear element, and for sitable vales of r e have (ignoring poers of v 2 ) b p 1,0 for ɛ>ɛ 2 d r (h 2,1 )= b pi i,0 for ɛ i >ɛ>ɛ i+1 b pi i,0 ± bpi 1 i 1,0 for ɛ = ɛ i. From these considerations e can dedce that or element x π ( X) represents the nontrivial permanent cycle x ɛ given by b 1,0 for ɛ>ɛ 2 x ɛ = b pi 1 i,0 ± b pi i+1,0 for ɛ = ɛ i b pi i+1,0 for ɛ i >ɛ>ɛ i+1 No since φ(b i,0,ɛ)=2 ɛ v 2 (2pi+1 2p 2) = 2 ɛ h ( p i+1 p 1 p 2 1 ),

10 DOUGLAS C. RAVENEL e have (8) 2 ɛ φ(x, ɛ) = 2p i 1 ( ) p 2 p 1 p 2 1 ɛp i 1 ( p i+1 p 1 p 2 1 ) for ɛ ɛ 2 for ɛ i ɛ ɛ i+1. Ths φ(x, ɛ) as a fnction of ɛ is pieceise linear and ɛφ(x, ɛ) is pieceise qadratic, its graph consisting of a contable collection of parabolic arcs. (This is not the reason for the term parabolic, hich ill be explained belo.) Close examination reveals that (9) (ɛ +2p 2) 2 ɛp(p 2 1) φ(x, ɛ) (ɛ +2p 2)2 (p +1) 2 4ɛp 2 (p 2, 1) i.e., the graph of the φ(x, ɛ) lies beteen to hyperbolas. The loer bond is obtained at the csp points ɛ i, and the line segments in the graph are each tangent to the pper hyperbola. From (9) e get (ɛ +2p 2) 2 p(p 2 1) ɛφ(x, ɛ) (ɛ +2p 2)2 (p +1) 2 4p 2 (p 2. 1) The pper and loer bonds are parabolas, bt in the relevant range (0 ɛ 1) they look very mch like straight lines. The graph of ɛφ(x, ɛ) is a contable collection of parabolic ares, each concave donard. Each arc is tangent to the pper line hile the loer line goes throgh the csp points, here adjacent arcs meet. Each arc has its maximm vale in the prescribed interval, hich is roghly [2p i, 2p 1 i ]; the i th arc achieves its maximm vale at ɛ = p 2 1 p i+1 p 1 1 p i 1 and the maximm vale is p i 1 (p 2 1) p i+1 p 1 so e get λ 1 (x) = p2 1 p 2. This vale of λ 1 (x) differs from the pper bond of the conjectre by a factor of p. Letx i π ( X) be an element representing b i,0 for ɛ>ɛ i+1. Then similar

comptations sho that and VARIATIONS ON THE TELESCOPE CONJECTURE 11 λ 1 (x i ) = p2 1 p i+1 λ 1 (x p 1 1 x p 1 2 x p 1 k ) = (p2 1)(p 1) p ( 1 p + 1 p 2 + + 1 ) p k = (p2 1)(p k 1) p k+1 1 < p2 1. p Local minima of ɛφ(x, ɛ) are achieved at the csp points ɛ i ; e find that hich gives and ɛ i φ(x, ɛ i )= 4(p 1)p2i (p +1)(p i 1) 2, 1) lim inf ɛφ(x, ɛ) =4(p ɛ 0+ p(p +1) µ(x) = (p +1)2. 4p 7. Terminology Finally, e ill discss the terminology sed in the last conjectre; it does not refer to the parabolic arcs described above. Consider the points on an Ext chart corresponding to b pi 1 i,0, for hich s =2pi 1 and t =2p i (p i 1). These points all lie on the parabola t s = p2 s 2 (10) (p +1)s, 2 along hich they are exponentially distribted. In order to find ɛφ(x, ɛ), look at the lines ith slope ɛ/ v 2 passing throgh these points, and choose ith one ith the highest s-intercept. This intercept is φ(x, ɛ), and ɛφ(x, ɛ) is v 2 times the prodct of intercept and the slope. A calcls exercise shos that the limiting vale of this prodct for lines tangent to the parabola is the parabola s focal length, and that λ 1 (x) is v 2 times this focal length. The nmber λ 2 (x) isthes-coordinate (in the (t s, s)-coordinate system) of the vertex of this parabola; hence the term displacement. We see in this case that λ 2 (x) =λ 1 (x)/(p 1). Replacing x by its prodct ith a linear element y old translate the parabola raise this qantity by the Adams-Novikov filtration of y. To find lim inf ɛ 0+ ɛφ(x, ɛ), consider the infinite convex polygon having these points as vertices. Local minima of ɛφ(x, ɛ) are achieved hen ɛ/ v 2 is the slope

12 DOUGLAS C. RAVENEL of one the edges the polygon, i.e., hen ɛ = ɛ i for some i 2. These lines are all tangent to the parabola (11) t s = p(p +1)2 s 2 (p +1)s. 8 The same calcls exercise shos that lim inf ɛ 0+ ɛφ(x, ɛ) is v 2 times the focal length of this parabola. It follos that µ(x) is the ratio beteen the to focal lengths, hence the term magnification. Projective geometry Here is another approach to this parabola. The convex polygon in the (t s, s)- plane is dal, in the sense of projective geometry, to the graph of the fnction φ(x, ɛ) (8)inthe(ɛ, φ)-plane. This dality is defined as follos. A nonvertical line in one plane has a slope and a y-intercept, i.e., the coordinate of its intersection ith the vertical axis. These to nmbers are (p to sitable scalar mltiplication) the coordinates of the dal point in the other plane. (We assme that the slope of the line is proportional to the horizontal coordinate of the dal point, and the intercept is proportional to the vertical one.) Conversely, given a point in one plane, each nonvertical line throgh it determines a point in the other plane and these points are collinear, so e get a line in the other plane dal to the original point. If e enlarge both affine planes to projective planes, then it is no longer necessary to exclde vertical lines. They are dal to points at infinity, and the line at infinity is dal to the point at infinity in the vertical direction. There is also a projective dality beteen crves. A crve in one plane has a collection of tangent lines, each of hich is dal to a point in the other plane. These points all lie on a ne crve, hich is defined to be the dal of the original crve. The tangent lines of the dal crve are dal to the points of the original crve. A parabola ith horizontal axis, sch as the one defined by (10), is dal to a hyperbola having the vertical axis as an assymptote, namely the pper one defined in (9). The loer hyperbola of (9) is dal to the parabola of (11). References [Dav] D.M.Davis.Comptingv 1 -periodic homotopy grops of spheres and some compact lie grops. To appear. [HS] M. J. Hopkins and J. H. Smith. Nilpotence and stable homotopy theory II. Sbmitted to Annals of Mathematics. [Mah82] M. E. Mahoald. The image of J in the EHP seqence. Annals of Mathematics, 116:65 112, 1982. [Mil81] H. R. Miller. On relations beteen Adams spectral seqences, ith an application to the stable homotopy of a Moore space. Jornal of Pre and Applied Algebra, 20:287 312, 1981.

VARIATIONS ON THE TELESCOPE CONJECTURE 13 [MRS] M. E. Mahoald, D. C. Ravenel, and P. Shick. The v 2 -periodic homotopy of a certain Thom complex. To appear. [Rava] D. C. Ravenel. A conterexample to the telescope conjectre. To appear. [Ravb] D. C. Ravenel. The parametrized Adams spectral seqence. To appear. [Rav84] D. C. Ravenel. Localization ith respect to certain periodic homology theories. American Jornal of Mathematics, 106:351 414, 1984. [Rav86] D. C. Ravenel. Complex Cobordism and Stable Homotopy Grops of Spheres. Academic Press, Ne York, 1986. [Rav92a] D. C. Ravenel. Nilpotence and periodicity in stable homotopy theory. Volme 128 of Annals of Mathematics Stdies, Princeton University Press, Princeton, 1992. [Rav92b] D. C. Ravenel. Progress report on the telescope conjectre. In N. Ray and G. Walker, editors, Adams Memorial Symposim on Algebraic Topology Volme 2, pages 1 21, Cambridge University Press, Cambridge, 1992. [Tod71] H. Toda. On spectra realizing exterior parts of the Steenrod algebra. Topology, 10:53 65, 1971. University of Rochester, Rochester, Ne York 14627 E-mail address: drav@troi.cc.rochester.ed