Differentials of the Adams Spectral Sequence and the Kervaire Invariant V

ISSN 1064Ð5624, Doklady Mathematics, 2009, Vol. 80, No. 1, pp. 573Ð576. © Pleiades Publishing, Ltd., 2009. Original Russian Text © V.A. Smirnov, 2009, published in Doklady Akademii Nauk, 2009, Vol. 427, No. 5, pp. 601Ð604.

MATHEMATICS

Differentials of the Adams Spectral Sequence and the Kervaire Invariant V. A. Smirnov Presented by Academician A.T. Fomenko March 20, 2009

Received April 2, 2009

DOI: 10.1134/S1064562409040310

This paper studies the differentials of the Adams coalgebra over an E∞-operad on the singular chain com- spectral sequence for stable homotopy groups of plex C*(X) of a simply connected topological space X spheres and solves the Kervaire invariant one problem i completely determines the weak homotopy type of this for n-manifolds with n = 2 Ð 2, where i > 6. space. In [8, 9], operad methods were used to describe The Kervaire invariant was defined by M. Kervaire the Adams spectral sequence for homotopy groups of for almost parallelizable smooth (4n + 2)-manifolds topological spaces and find its differentials. [1], where it was used to construct an example of a PL To describe the Adams spectral sequence, we use the 10-manifold not admitting a smooth structure. differential graded algebra Λ [10]. As a graded algebra, Browder [2] proved that the Kervaire invariant van- λ ≥ it is generated by elements i of dimension i 0 and the ishes if n + 2 is not a power of 2. Barratt et al. [3] showed relations that the Kervaire invariant equals 1 for some n-manifolds in the case of n = 2i – 2, where i = 2, 3, 4, 5, 6. Since ⎛⎞ λ λ njÐ1Ð λ λ then, the question of whether the Kervaire invariant 2in++1 i = ∑⎜⎟2ij++1 injÐ+ . i ⎝⎠ equals 1 for n-manifolds with n = 2 Ð 2 and i > 6 has j j remained open. Λ In the language of spectral sequences [4], this ques- Thus, as a graded module, the algebra is generated by elements of the form λ λ …λ , in which i ≤ 2i , tion can be reformulated as follows: Do the elements i1 i2 ik 1 2 2 2 i ≤ 2i , …, i ≤ 2i . Such sequences i , i , …, i are hn of the term E of the Adams spectral sequence for 2 3 k – 1 k 1 2 k stable homotopy groups of spheres survive up to the said to be admissible. term E∞? On the algebra Λ, a differential d is defined; it acts λ In this paper, we suggest an algebraic approach to on the generators i as answering this question, which is based on the homo- ⎛⎞ topy theory of coalgebras over operads in the category ()λ ikÐ λ λ d i = ∑⎜⎟k 1Ð ikÐ . of chain complexes; the main results of this theory were ⎝⎠k reported at M.M. Postnikov’s algebraic topology semi- k nar. Λ 1 1 Recall that the notion of an operad was introduced The algebra is isomorphic to the term E = {E pq, } by May [5] in order to describe the structure on iterated of the Adams spectral sequence for stable homotopy loop spaces. In [6], this notion was used to describe the groups of spheres, and its homology with respect to the 2 algebraic structure on the cochain complex of a topo- differential d is isomorphic to the term E = {E , } of logical space. In particular, it was proved that the singu- 2 pq this spectral sequence. lar cochain complex C*(X) of a topological space X carries the structure of an algebra over an E∞-operad E. In Direct calculations show that the unique cycles in 1 [7], the homotopy theory of coalgebras over operads the term E , are the elements λ n . The correspond- was constructed. It was proved that the structure of a 1 * 2 1Ð ing homology classes are denoted by hn. They generate E2 the term 1, * . Moscow State Pedagogical University, 2 ul. Malaya Pirogovskaya 1, Moscow, 119882 Russia; The term E2, is generated by the products hnhm, ≤ * e-mail: [email protected] where n m, and the relations hnhn + 1 = 0.

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The algebra Λ makes it possible to calculate not algebra Λ˜ is isomorphic to the term E∞ of the Adams only stable homotopy groups of spheres but also unsta- spectral sequence for stable homotopy groups of ble homotopy groups of “nice” topological spaces in spheres. If X is a nice topological space (that is, H∗(X) the MasseyÐPeterson sense [11]. is the exterior algebra over the graded module M Recall that the homology of a nice topological space over the Steenrod algebra), then, on the module M, a X is the exterior algebra generated by a graded module Λ˜ ϕ˜ M over the Steenrod algebra. Examples of nice spaces -comodule structure can be defined so that the ˜ are the sphere, classical topological groups, etc. homology of the complex Λ ×ϕ˜ M is isomorphic to the For a graded module M, let Λ × M denote the sub- term E∞ of the Adams spectral sequence of homotopy module of Λ ⊗ M generated by the elements groups of X. λ λ …λ ⊗ x , where i , i , …, i form an admissible i1 i2 ik n 1 2 k A proof of this theorem is contained in [12]. Here, sequence and ik < n. we only describe its main stages. Let E denote the We say that a module M is a Λ-comodule if an oper- comonad in the category of chain complexes corre- ation ϕ: M → Λ × M satisfying the relation sponding to the E∞-operad E. The singular chain complex C (X) of the topological space X is a coalgebra d()ϕ +0 ϕ∪ ϕ= ∗ E X is defined, where ϕ ∪ ϕ = (γ × 1)(ϕ × 1)ϕ: M → Λ × M over the comonad . The homotopy groups of can be and γ: Λ ⊗ Λ → Λ is multiplication in the algebra Λ. calculated by using the co-B-construction Λ 2 For a -comodule M, we can define a differential dϕ FE():, C ()X C ()X →→EC()()X E ()C ()X →… . on the module Λ × M by setting * * * * Let us pass from this co-B-construction to its homol- dϕ()yx⊗ = dy()⊗ϕx + yx⊗∩ , ogy. The homology E of the comonad E carries an * where ϕ∩ = (γ × 1)(1 × ϕ): Λ × M → Λ × M. We denote A∞-comonad structure consisting of operations the corresponding differential graded module by Λ ×ϕ M. n δn: E → E 2+ , n ≥ 0, If X is a nice topological space and H∗(X) is the * * exterior algebra generated by the graded module M, increasing dimension by n. The operations δn commute then, on M, the Λ-comodule structure with the suspension S: SE → E . ϕ() λ ⊗ () * * xn = ∑ i 1Ð Sqi xn The homology H∗(X) of the topological space X car- i ries an A∞-coalgebra structure over the A∞-comonad 1 is defined. The term E (X) of the Adams spectral E , which consists of operations sequence for homotopy groups of the topological space * Λ × n n 1+ X is isomorphic to the module M, and the homology τ : H ()X → E ()H ()X , n ≥ 0, of the differential graded module Λ ×ϕ M is isomorphic * * * to the term E2(X) of this spectral sequence. In particular, increasing dimension by n and commuting with the sus- the homology of the complex Λ × H∗(Sn), where Sn is pension. These structures determine a new differential on the co-B-construction the n-sphere, is isomorphic to the term E2 of the Adams spectral sequence for homotopy groups of the sphere Sn. FE(), H ()X : H ()X → E ()H ()X * * * * * Consider the special orthogonal group SO. Its 2 → E ()…H ()X → , homology is the exterior algebra over a module isomor- * * phic to the homology module of the real projective ∞ the homology with respect to this differential is isomor- space RP . The Λ-comodule structure is defined on the phic to the term E∞ of the Adams spectral sequence for ∈ ∞ generators xn H∗(RP ) by homotopy groups of X. ⎛⎞ Recall that the homology H∗((E C∗(X))) = ϕ() ⎜⎟nkÐ λ ⊗ xn = ∑ k 1Ð xnkÐ . E (H∗(X)) is isomorphic to the exterior algebra over ⎝⎠k * k the graded module (H*(X)), which is generated by ∞ The homology of the complex Λ ×ϕ H∗(RP ) is iso- elements of the form e e …e ⊗ x of dimension j1 j2 jk n 2 k k – 1 ∈ ≤ ≤ ≤ morphic to the term E of the Adams spectral sequence 2 n – 2 jk – … – j1, where xn Hn(X) and 0 j1 j2 ≤ ≤ for homotopy groups of the space SO. … jk n. Λ ˜ Theorem 1. On the algebra , a new differential d The suspension S: S → is defined by and a new multiplication γ˜ can be defined so that the Se()e …e = e e …e . homology of the corresponding differential graded j1 j2 jk j1 1+ j2 1+ jk 1+

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The A∞-comonad structure on E induces an A∞- Note that multiplication γ in the algebra Λ is subject * comonad structure on , which consists of operations to a perturbation too. The value of the perturbed multi- n n + 2 γ Λ ⊗ Λ → Λ λ ⊗ λ δ : → , where n ≥ 0, increasing dimension by n plication ˜ : at an element j i coincides γ λ ⊗ λ λ λ ≤ and commuting with the suspension. with ( j i) = j i provided that j 2i. In the case of γ λ ⊗ λ If X is a nice topological space, then the structure of j > 2i, the value ˜ ( j i) is obtained by adding some λ λ λ γ λ ⊗ λ an A∞-coalgebra over the A∞-comonad E on the new elements i i … i , where k > 2, to ( j i). * 1 2 k homology H∗(X) induces the structure of an A∞-coalge- Since the operations δn commute with the suspen- bra over the A∞-comonad on the module M, which sion, it follows that the value of the perturbed multipli- consists of operations τn: M → n + 1(M), where n ≥ 0, cation γ˜ at elements of the form e ⊗ e , where p ≤ q, increasing dimension by n and commuting with the sus- p q consists of elements of the form e j e j …e j with j1 > pension. 1 2 k ≥ j2 > jk p. Passing from elements of the form ej to ele- In this case, the co-B-construction F(,E H∗(X)) λ * ments of the form i, we see that, for j > 2i, the value can be replaced by the co-B-construction γ˜ (λ ⊗ λ ) consists of elements λ λ …λ with j i i1 i2 ik 2 F(), M : M →→()M ()M → …, ik < j – i. whose homology is isomorphic to the term E∞ of the Now, let us formulate the main theorem. Adams spectral sequence for homotopy groups of X as Theorem 2. The elements h2 , where n ≥ 1, of the well. n Adams spectral sequence for stable homotopy groups δ0 → 2 ∞ Note that the operation : determines a of spheres survive up to the term E . comonad structure on . If M is treated as the trivial - Proof. There is a J-homomorphism J: π∗(SO) → σ∗ coalgebra, then the homology of the corresponding co- π B-construction F(, M) is isomorphic to the graded mod- from the homotopy groups ∗(SO) to the stable homo- ule L generated by elements of the form e e …e ⊗ x topy groups of spheres σ∗ [10]. On the terms E1 of the j1 j2 jk n k k – 1 ∈ corresponding spectral sequences, it has the form of dimension 2 n – i1 – 2i2 – … – 2 ik – k, where xn k – 2 k – 1 ∞ M, j1 > j2 > … > jk, and j1 + j2 + … + 2 jk < 2 n. Λ × ()Λ→ J: H* RP Let us rewrite these elements in the form and is deﬁned by J(…λ λ λ ⊗ x ) = λ λ …λ λ . λ λ …λ ⊗ x , where i = n – j – 1, i = 2n – j – i1 i2 ik n i1 i2 ik n i1 i2 ik n k k k – 1 k k – 1 k – 2 jk – 1 – 1, …, i1 = 2 n – 2 jk – … – j2 – 1. For these Let Λ˜ ' denote the subcomplex of Λ˜ generated by ≤ ≤ elements, we have ik < n, ik – 1 2ik, …, i1 2i2. Thus, λ λ λ the elements i i … i for which i1i2…ik is an admis- the graded module L is isomorphic to the graded mod- 1 2 k ule Λ × M. sible sequence and ik – 1 < ik. It is easy to see that the It follows from the perturbation theory of differen- homomorphism J induces the isomorphism of chain complexes tials [13] that the remaining operations δn: → n + 2 ∞ ≥ ˜ Λ Λ˜ × ()Λ≅ ˜ with n 1 induce a differential d on , and the struc- ϕ H* RP '. ture of an A∞-coalgebra over the A∞-comonad on the Recall that the homotopy groups π∗(SO) vanish in ˜ module M induces a Λ -comodule structure ϕ˜ : M → the dimensions 2n + 1 – 3 for n > 1. Therefore, the homol- Λ˜ × M on M. ogy of the complex Λ˜ ' vanishes as well in these dimen- The homology of the corresponding complex sions. Λ˜ × ∞ ϕ˜ M is isomorphic to the term E of the Adams Let us prove that the value of the differential d˜ at spectral sequence for homotopy groups of X. This com- ˜ λ n λ n belongs to Λ' . We have pletes the proof of the theorem. 2 1Ð 2 1Ð Λ˜ ˜ ˜ In particular, the homology of the complex is iso- d()γλ n λ n = ˜ ()d()λλ n ⊗ n 2 1Ð 2 1Ð 2 1Ð 2 1Ð morphic to the term E∞ of the Adams spectral sequence ˜ + γ˜ ()λ n ⊗ d()λ n . for stable homotopy groups of spheres, the homology 2 1Ð 2 1Ð Λ˜ × n ∞ of the complex H∗(S ) is isomorphic to the term E It is easy to see that the ﬁrst term of the sum on the of the Adams spectral sequence for homotopy groups of right-hand side belongs to Λ˜ ' . Let us show that the sec- n Λ˜ × the sphere S , and the homology of the complex ϕ˜ ˜ ˜ ond term belongs to Λ' also. Indeed, d ()λ n belongs H∗(RP∞) is isomorphic to the term E∞ of the Adams 2 1Ð spectral sequence for homotopy groups of the space SO. to Λ˜ ' and is the sum of elements of the form

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λ λ …λ , where k > 2 and i + i + … + i = 2n Ð 2. REFERENCES i1 i2 ik 1 2 k n − ≤ λ λ λ λ If 2 1 2i1, then the elements n i i … i 1. M. Kervaire, Comment. Math. Helv. 34, 257Ð270 2 1Ð 1 2 k (1960). ˜ n belong to Λ' . If 2 – 1 > 2i , then, multiplying λ n by 1 2 1Ð 2. W. Browder, Ann. Math. 90, 157Ð186 (1969). λ , we obtain products λ λ …λ in which the 3. M. G. Barratt, J. D. S. Jones, and M. E. Mahowald, i1 j1 j2 jl n J. London Math. Soc. 30 (3), 533Ð550 (1984). dimension of the last factor jl is at most 2 – i1 Ð 2. If j > 2i , then, multiplying this factor λ by λ , we 4. J. F. Adams, Comment. Math. Helv. 32, 180Ð214 (1958). l 2 jl i2 obtain products in which the dimension of the last ele- 5. J. P. May, The Geometry of Iterated Loop Spaces n (Springer-Verlag, Berlin, 1972). ment is at most 2 – i1 – i2 Ð 3. Continuing, we see that 6. V. A. Smirnov, Mat. Sb. 115 (1), 146Ð158 (1981). the dimension of the element preceding λ is at most ik n 7. V. A. Smirnov, Izv. Akad. Nauk SSSR 46 (6), 1302Ð1321 2 – i1 – … – ik – 1 – (k – 1) = ik – k + 2, which is less (1985). than i ; therefore, the product γ˜ (λ n ⊗ λ λ …)λ k 2 1Ð i1 i2 ik 8. V. A. Smirnov, Izv. Akad. Nauk, Ser. Mat. 66 (5), 193Ð 224 (2002). belongs to Λ˜ ' . 9. V. A. Smirnov, Mat. Zametki 84, 763Ð771 (2008). ˜ ˜ Thus, the element d ()λ n λ n is a cycle in Λ' of 2 1Ð 2 1Ð 10. G. W. Whitehead, Recent Advances in Homotopy Theory dimension 2n + 1 Ð 3. The vanishing of the homology of (Am. Math. Soc., Providence, R.I., 1970; Mir, Moscow, 1974). the complex Λ˜ ' in these dimensions implies the exist- 11. W. S. Massey and F. P. Peterson, The Mod 2 Cohomology ence of an element b n 1+ belonging to the complex 2 2Ð Structure of Certain Fibre Spaces (Am. Math. Soc., ˜ ˜ ˜ Providence, R.I., 1967). Λ' for which d ()b n 1+ = d (λ n λ n ). Therefore, 2 2Ð 2 1Ð 2 1Ð 12. V. A. Smirnov, Simplicial and Operad Methods in Alge- the element λ n λ n + b n 1+ is a cycle in the com- braic Topology (Am. Math. Soc., Providence, R.I., 2 1Ð 2 1Ð 2 2Ð 2001). Λ˜ 2 ≥ plex , i.e., the elements hn with n 1 survive up to 13. V. K. Gugenheim, L. A. Lambe, and J. D. Stasheff, Illi- the term E∞. nois J. Math. 35 (3), 357Ð363 (1991).

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