The Finiteness Conjecture

Transcription

1 Robert Bruner Department of Mathematics Wayne State University The Kervaire invariant and stable homotopy theory ICMS Edinburgh, Scotland April 2011 Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 1 / 59

2 Outline 1 The Finiteness Conjecture The Conjecture Squaring operations Iteration Iteration improved Jones s Kervaire class in the 30 stem No more such examples Finiteness conjecture revisited 2 Equivariant perspectives The Bredon and Mahowald root invariants Easy equivariant proofs Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 2 / 59

3 The Conjecture We will focus entirely on the prime 2 in this talk. Conjecture (The Finiteness Conjecture) A Sq 0 family {x, Sq 0 (x), Sq 0 Sq 0 (x),..., (Sq 0 ) n (x),...} in Ext A (F 2, F 2 ) = π S detects only a finite number of non-zero homotopy classes. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 5 / 59

4 The Conjecture Examples {h 0, h 1, h 2, h 3 } detect the Hopf invariant one maps 2, η, ν, and σ. All higher members of this family die at E 2 by the differential d 2 (h n+1 ) = h 0 h 2 n. {h0 2, h2 1, h2 2, h2 3, h2 4, h2 5 } and perhaps h2 6 detect Kervaire invariant one maps. Hill, Hopkins and Ravenel have shown that the remaining hn 2 cannot be permanent cycles, but we do not yet know the differentials which might kill them. {c 0, c 1 } detect ɛ π 8 and ɛ 1 π 19 while d 2 c i = h 0 f i 1 for i 2. d 2 f 0 = h 2 0 e 0, f 1 survives to at least E 5, and d 3 f i = h 1 y i 1 for i 2. d 2 e 0 = c 2 0 = h2 1 d 0, d 3 e 1 = h 1 t 0 and d 2 e i = h 0 x i 1 for i 2. d 2 y i = h 0 h i+3 r i for all i 0 (note y 1 = h 4 Q 3 ) Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 6 / 59

5 The Conjecture Remark Minami called this conjecture The new Doomsday Conjecture. His papers The iterated transfer analogue of the new doomsday conjecture, Trans. AMS 351 (1999) and The Adams spectral sequence and the triple transfer, Am J. Math. 117 (1995) provide some evidence, using the transfer, that it is true. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 7 / 59

6 The Conjecture All members of a Sq 0 -family lie in the same Adams filtration Ext s,. Sq 0 is a ring homomorphism by the Cartan formula. The original Doomsday Conjecture was that each filtration of the Adams spectral sequence detects only a finite number of homotopy classes. That was definitively refuted by Mahowald s η j family, detected by the elements h 1 h j Ext 2,2j +2. However, these do not form a Sq 0 -family, since Sq 0 (h 1 h j ) = h 2 h j+1, Sq 0 (h 2 h j+1 ) = h 3 h j+2,... With a finite number of low dimensional exceptions, the only elements in Adams filtration 2 which could be non-zero permanent cycles are the hj 2 and the h 1 h j. Thus, the Sq 0 -families generated by the h 1 h j obey the finiteness conjecture. The generic Adams differential is d 2 Q i x = h 0 Q i 1 x if i is even. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 8 / 59

7 The Conjecture Goal To explain why I have always suspected that the Finiteness Conjecture is true. Goal To show how these methods give easy proofs of some of the results already mentioned this week. To gain some perspective on the Finiteness Conjecture, some Corollaries are: There are only a finite number of Hopf invariant one maps. There are only a finite number of Kervaire invariant one maps. Red shift (in K-theory and homotopy theory) is complicated. So, it is unlikely that we will have a proof in the near future. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 9 / 59

8 Squaring operations Squaring operations The cohomology of a cocommutative Hopf algebra, such as the Steenrod algebra, has natural operations Sq i : Ext s,t A (F 2, F 2 ) Ext s+i,2t A (F 2, F 2 ) for 0 i s in the cohomological indexing, or Q i : Ext s,t A (F 2, F 2 ) Ext s+t i,2t A (F 2, F 2 ) for t s i t in the homological indexing. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 11 / 59

9 The Finiteness Conjecture Squaring operations Cohomological indexing: 2s Sq s x 2s 1 Sq s 1 x s x Sq 0 x n 2n 2n + 1 2n + s Sq i : Ext s,t Ext s+i,2t (n = t s) Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 12 / 59

10 The Finiteness Conjecture Squaring operations Homological indexing: 2s Q n x 2s 1 Q n+1 x s x Q t x n 2n 2n + 1 2n + s Q i : Ext s,t Ext s+t i,2t (n = t s) Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 13 / 59

11 Squaring operations S-algebra structure of the sphere The product µ : S S S factors through the homotopy orbits Some notation: µ S S S ξ D 2 S := (S S) hc2 For G Σ r, D G X := (X r ) hg Skeleta: D i 2 X := S i + C2 X X and D i G X := EG i + G X r Observe that D2 i S n = Σ n RPn n+i, where Pn k = RP k /RP n 1, the stunted projective space with cells in dimensions n through k. P n = P n. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 14 / 59

12 Squaring operations Homotopy operations S n x S D G S n D G x D G S ξ S Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 15 / 59

13 Squaring operations Homotopy operations S n x S D G S n D G x D G S ξ S α α (x) S k Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 16 / 59

14 The Finiteness Conjecture Squaring operations Cup-i operations We call the operation cup-i S n x S D 2 S n D2x D 2 S ξ S i i (x) S 2n+i if i π 2n+i D 2 S n π 2n+i Σ n P n gen H 2n+i D 2 S n Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 17 / 59

15 Squaring operations Properties 0 (x) = x 2 and always exists. i : π n π 2n+i is detected by Q n+i in Ext Each cell of D 2 S n either defines a i operation or a relation between lower operations. For example, 1 : π n π 2n+1 exists iff n is even. If n is odd then the 2n + 1 cell of D 2 S n = Σ n P n instead gives a null-homotopy of 2x 2. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 18 / 59

16 Squaring operations Cup-1 of 2 is η D 2 S D22 D 2 S ξ S 1 η S 1 This is detected by Sq 0 (h 0 ) = h 1 in Ext. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 19 / 59

17 Squaring operations Cup-1 of η is not defined D 2 S 1 D 2η D 2 S ξ S 1 S 3 However, we do have Sq 0 (h 1 ) = h 2 in Ext. Restricting to the 3-skeleton, Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 20 / 59

18 Squaring operations D2 1S 1 D 2η D2 1S ξ S ν top S 3 The attaching map of the 3-cell of ΣP 1 has degree 2, and this gives an Adams spectral sequence differential d 2 (h 2 ) = h 0 h1 2 = 0, and there are no possible higher differentials, allowing ν to exist. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 21 / 59

19 Squaring operations Similarly, D2 1S 3 D2ν D2 1S ξ S σ top S 7 Again, the attaching map has degree 2, and this gives d 2 (h 3 ) = h 0 h 2 2 = 0, and there are no possible higher differentials, allowing σ to exist as well. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 22 / 59

20 Squaring operations After this, the differential d 2 (h n+1 ) = h 0 h 2 n 0, and no higher Hopf maps exist. In this sense, η must exist, while ν and σ are gifts, or low dimensional accidents. The 15 cell carrying h 4 is a null-homotopy of 2σ 2, showing that 2θ 3 = 0. For higher n, we don t get the implication 2θ n = 0 from the differential d 2 (h n+1 ) = h 0 h 2 n, though, because h n was not a homotopy class to start with and the story is a bit more complicated. The boundary of the cell carrying h n decomposes into a part carrying h 0 h 2 n and a part carrying operations on h 0 h 2 n 1, effectively setting 2θ n equal to higher Adams filtration elements which we must analyze. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 23 / 59

21 Squaring operations Operations on relations If 2x = 0 we can extend and operate on the extension as before S n 2 e n+1 x S D G (S n 2 e n+1 ) D G x D G S ξ S Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 24 / 59

22 Squaring operations Operations on relations If 2x = 0 we can extend and operate on the extension as before S n 2 e n+1 x S D G (S n 2 e n+1 D G ) x D G S ξ S α α (x) S k Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 25 / 59

23 Squaring operations To talk about π D 2 (X ) and H D 2 (X ): for a space X, Σ QX r Σ D Σr X (suppress Σ henceforth) H QX is the free Dyer-Lashof module on H X H D r X is the summand of weight r, where wt(h X ) = 1 wt(ab) = wt(a) + wt(b) wt(q i (a)) = 2wt(a) the Nishida relations tell us the A-module structure of H QX. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 26 / 59

24 Squaring operations As an application, Theorem If θ n 1 exists, has order 2 and square 0, then θ n exists and has order 2. Proof: Let N = 2 n 2 and let X = S N 2 e N+1 and let x H N X, y H N+1 X be the generators. Let θ : X S be an extension of θ n 1 by a nullhomotopy of 2θ n 1. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 27 / 59

25 Squaring operations The bottom 4 dimensions of H D 2 X and the attaching maps are shown at left, together with their images under ξd 2 θ n 1 and the detecting elements in the Adams spectral sequence on the right. 2N 2N + 1 x 2 θn 1 2 hn 1 4 = 0 Q N+1 x 1 (θ n 1 ) Q N+1 (h 2 n 1 ) = 0 xy - h 2 n 1 h n = 0 2N + 2 2N + 3 Q N+2 x θ n y 2 θ n hn 2 hn 2 Q N+3 x - 0 Q N+2 y - 0 The assumption that θn 1 2 = 0 means that D 2X θ S factors through the quotient by the bottom cell. The cell y 2 is then unattached and gives θ n, while the cell Q N+2 y gives a nullhomotopy of 2θ n. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 28 / 59

26 Squaring operations Other operations like h 1 1 There are many other operations than the i. For example, if n = 1 (mod 4), there is an indecomposable homotopy operation h 1 1 detected by h 1 Q n+1 in the Adams spectral sequence. This operation obeys the relation 2 h 1 1 (x) = η2 x 2. (Draw Picture.) Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 29 / 59

27 Iteration Iteration Let x 0 := x and x n = Sq 0 (x n 1 ) be a Sq 0 -family. Suppose x : S t s S has Adams filtration s. Then x 1 lives on the top cell of Σ t s Pt s, t so lies in the 2t s stem. Similarly, x 2 lives on the top cell of Σ 2t s P2t s, so lies in the 4t s stem. In general, x i+1 is carried by the top cell of Σ 2i t s P 2i t 2 i t s. Thus, the stems in which the x i lie are converging to s 2-adically, while the cell of projective space on which x i is carried is converging to 0 2-adically. The 0 cell of P is attached to every cell below it: Sq i (x i ) = x 0, so that for large i, x i will only survive if all s obstructions vanish: h 0 Sq 1 (x i 1 ), h 1 Sq 2 (x i 1 ), h 1, h 0, Sq 3 (x i 1 ), h 2 Sq 4 (x i 1 ),..., down to the obstruction involving Sq s (x i 1 ) = xi 1 2. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 31 / 59

28 Iteration S t s x 0 S D2 ss t s D 2 x 0 S x 1 S 2t s D2 ss 2t s D 2 x 1 S x 2 S 4t s Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 32 / 59

29 Iteration D2 ss t s D 2 x 0 S x 1 S 2t s D2 ss 2t s D 2 x 1 S x 2 S 4t s D2 ss 4t s D 2 x 2 S x 3 S 8t s etc. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 33 / 59

30 The Finiteness Conjecture Iteration improved Iteration improved Rather than apply D 2 only to x i to get x i+1, we could apply it to the whole triangle above, and use the natural map D 2 D 2 D 4 : D2 sds 2 S t s D2 ss 2t s top S 4t s S 4t s S 4t s k D4 3s S t s S Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 35 / 59

31 The Finiteness Conjecture Jones s Kervaire class in the 30 stem The classes we could operate upon to reach the Kervaire class in dimension 30 are h0 2 Q 0 h 0 Q 1 Q 2 h1 2 Q 4 h2 2 Q 8 h3 2 Q 16 h4 2 Q 1 Q 3 Q 7 Q 15 h 1 h 2 h 3 h 4 Q 2 Q 4 Q 8 The differential d 2 (h 4 ) = h 0 h 2 3 means that we cannot use h 4 in any simple way. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 37 / 59

32 The Finiteness Conjecture Jones s Kervaire class in the 30 stem h0 2 Q 0 h 0 Q 1 Q 2 h1 2 Q 4 h2 2 Q 8 h3 2 Q 16 h4 2 Q 1 Q 3 Q 7 Q 15 h 1 h 2 h 3 h 4 Q 2 Q 4 Q 8 Next simplest is Q 16 (h3 2) or 2(θ 3 ). This lives on the top (i.e., 30) cell of D2 2S 14 = Σ 14 P14 16: This shows that θ 4 enforces the relation 2 1 (θ 3 ) + ηθ3 2 = 0. For it to be a homotopy class rather than a null-homotopy, this relation would have to have already been true before θ 4 arrived to enforce it. This sounds like metaphysics, but is really an extension problem. Precisely, Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 38 / 59

33 The Finiteness Conjecture Jones s Kervaire class in the 30 stem Consider an Adams resolution of S. S 30 e 30 S 29 Σ 14 P Σ 14 P S 28 Y 2 Y 3 Y 4 S Y 1 The 30-cell carrying θ 4 causes 2 1 (θ 3 ) + ηθ 2 3 to be 0 in π Y 2. It lives naturally in π Y 3, and if it were 0 there, it map would factor through a point and θ 4 would exist. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 39 / 59

34 The Finiteness Conjecture Jones s Kervaire class in the 30 stem h0 2 Q 0 h 0 Q 1 Q 2 h1 2 Q 4 h2 2 Q 8 h3 2 Q 16 h4 2 Q 1 Q 3 Q 7 Q 15 h 1 h 2 h 3 h 4 Q 2 Q 4 Q 8 Next consider Q 16 Q 7 (h 3 ). Write classes in H D 4 S 7 as follows: Q i for the homology class Q i ι 7, Q i Q j for Q i Q j ι 7, and Q i Q j, etc., for their products. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 40 / 59

35 Jones s Kervaire class in the 30 stem Here are the bottom few cells of D 2 4 S ι 4 7 θ 2 3 h 4 3 = 0 5 ι 2 7 Q8 - h 2 3 h 4 = 0 4 ι 2 7 Q9 - h 2 3 Q9 h 3 = 0 4 Q 8 Q 8 θ 4 h ι 2 7 Q10 - h3 2Q10 h 3 = 0 6 Q 8 Q 9 - h 4 Q 9 h 3 = 0 3 Q 16 Q 8 - h 5 1 Q 8 Q 8 + ι 2 7 Q9 is spherical, the operation it represents takes σ to θ 4, Q 16 Q 8 + ι 2 7 Q10 is attached by a map of degree 2 to Q 8 Q 8 + ι 2 7 Q9, so 2θ 4 = 0. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 41 / 59

36 Jones s Kervaire class in the 30 stem Manifold realization D 4 S 7 = T (7ρ 4 ), the Thom complex of 7ρ 4, Thom isomorphism Φ : H D 4 S 7 H BΣ 4, the weight 4 summand of H QS 0. Extended powers of manifolds and the Adams spectral sequence Contemp. Math., 271 (1999), gives a dictionary {x H D r S n } Ψ {M f BΣ r f [M] = Φ(x)} so that x is spherical iff f (nρ r ) = ν M and Ψ(Q 8 Q 8 + ι 2 7 Q10 ) is Jones (α : π n π k ) (N Ψ(h(α)) Σr N r ). S 1 S 1 #RP 2 BΣ 4 Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 42 / 59

37 No more such examples What have we done? We couldn t operate successfully on θ 3 = σ 2 : S 14 S but by backing up a step, to σ : S 7 S we were successful: D 2 S 14 D 4 S 7 D4σ D 4 S S J θ 4 S 30 Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 44 / 59

38 No more such examples This suggests various strategies for higher Kervaire elements. Look for D 8 S 7 D8σ D 8 S S? θ 5 S 62 D 16 S 7 D16σ D 16 S? θ 6 S 126 S but [Jones] there are no spherical classes which contain the classes needed to produce these elements. In fact, the attaching maps can be reduced to η, attaching to a cell carrying θ4 2 or θ2 5, resp., but no further. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 45 / 59

39 The Finiteness Conjecture No more such examples There is another possibility. I thought for a bit that there might be a nice reverse symmetry, and we d find but and finally D 16 S 3 D16ν D 16 S S B θ 5 S 62 D 32 S 3 D32ν D 32 S θ 6 S 126 S D 64 S 1 D 64η D 64 S S θ 6 B S 126 Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 46 / 59

40 No more such examples But these meet exactly the same obstructions. (I am fairly certain.) This strongly suggests ηθ 2 n is the obstruction to θ n+1 ηθ 2 n is accidentally 0 in a couple more cases to allow the last few Kervaire classes, but ηθ if θ 6 exists, or ηθ if not. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 47 / 59

41 Finiteness conjecture revisited As we iterate Sq 0, we find that the number of obstructions which must cancel grows. Don Davis results, saying that the h i act monomorphically on initial segments of Ext in a range growing with i suggest that the stable obstruction h 0 Sq 1 + h 1 Sq 2 + < h 1, h 0, Sq 3 > +h 2 Sq 4 + will be nonzero generically. Nishida s theorem tells us that the bottom cells of these large extended powers must map trivially, and it seems likely that this will extend some distance up from the bottom, setting up a race between nilpotence at the bottom and Sq 0 at the top of the large truncated extended powers. The root invariant is often detected by Sq 0 in the range we have seen, but if the Finiteness Conjecture holds, then this process must continually be interrupted as we iterate the root invariant. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 49 / 59

42 Equivariant perspectives The Bredon and Mahowald root invariants The Bredon and Mahowald root invariants Because of the connection with the root invariant, I want to show you the equivariant version of the root invariant. Definition Extend x : S n S to x : S n+kτ S with k maximal. The Bredon root invariant, B(x) is then the underlying non-equivariant map B(x) = U(x) : S n+k S. The cofiber sequence C 2+ S S τ smashed with S n+kτ shows that the obstruction to extending one further is the composite C 2+ S n+kτ S n+kτ x S which is the adjoint of U(x), so B(x) is always nonzero. As with Mahowald s root invariant, it is clearly a coset. Restricting x to fixed points gives x. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 52 / 59

43 Equivariant perspectives The Bredon and Mahowald root invariants Theorem (Greenlees and B) The Bredon root invariant equals the Mahowald root invariant. See The Bredon-Löffler conjecture Experiment. Math. 4 (1995), no. 4, Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 53 / 59

44 Equivariant perspectives Easy equivariant proofs Theorem The root invariant at least doubles the stem. Proof: S n x S x x S n S n Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 55 / 59

45 Equivariant perspectives Easy equivariant proofs Theorem B(2) = η, B(η) = ν, B(ν) = σ. Proof: If D is one the the division algebras R, C or H then its double D has an involution whose fixed point set is D. The Hopf construction on D has the Hopf construction on D as its fixed points, and this extension is maximal. Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 56 / 59

46 Equivariant perspectives Easy equivariant proofs Cartan formula Theorem Let x i π ni S 0 and B(x i ) π ni +k i S 0, for i = 1, 2. Let k = k 1 + k 2 and let i : S k 1 P k 1 be the inclusion of the bottom cell of the stunted projective space P k 1. If i (B(x 1 )B(x 2 )) 0 then B(x 1 )B(x 2 ) B(x 1 x 2 ). If i (B(x 1 )B(x 2 )) = 0 then B(x 1 x 2 ) lies in a higher stem than does B(x 1 )B(x 2 ). Proof: Certainly, the smash product of extensions of x 1 and x 2 is an extension of x 1 x 2. The condition determines whether or not it is maximal. See Some remarks on the root invariant Stable and unstable homotopy (Toronto, ON, 1996), Fields Inst. Commun Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 57 / 59

47 Equivariant perspectives Easy equivariant proofs Thank you Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 58 / 59

48 Equivariant perspectives Easy equivariant proofs Robert Bruner (Wayne State University) The Finiteness Conjecture ICMS Edinburgh 59 / 59