On the boundaries of highly connected, almost closed manifolds

Abstract

Theorem 1.4. (Conjecture of Galatius and Randal-Williams) Let MO⟨4n⟩ denote the Thom spectrum of the canonical map τ ⩾4n BO −! BO, where τ ⩾4n BO denotes the (4n−1)-connected cover of BO. For all n>31, the unit mapis surjective, with kernel exactly the image of the J-homomorphism π 8n−1 O −! π 8n−1 S .We label Theorem 1.4 a conjecture of Galatius and Randal-Williams since it is, when n>31, equivalent to Conjectures A and B of their work [35]. Theorem 1.4 allows us to improve the bound k>232 in the d=0 case of Theorem 1.1. For details, see Theorem 8.6.Remark 1.5. Much of Theorem 1.4 is classical: the surjectivity statement follows from surgery as in [50, Theorem 6.6], while the Pontryagin-Thom correspondence guarantees that the image of the J-homomorphism is contained in the kernel of the unit map π 8n−1 S!π 8n−1 MO⟨4n⟩. The difficult point is to prove that the kernel of this unit map contains only the image of J.A priori, there could be additional elements in this kernel, and the concern has a geometric interpretation. Let Σ Q ∈Θ 8n−1 denote the boundary of the manifold obtained by plumbing together two copies of the 4n-dimensional linear disk bundle over S 4n that generates the image of π 4n BSO(4n−1) in π 4n BSO(4n). Theorem 1.4 is equivalent to the claim that, for n>31, the class [Σ Q ]∈coker(J) 8n−1 is trivial [90, Lemma 10.3].Our proof of Theorem 1.4 follows a general strategy due to Stolz [90], which he applied to prove some cases of Theorem 1.1. For each prime number p we compute a lower bound on the HF p -Adams filtrations of classes in the kernel of the unit mapOur lower bound is given in Theorem 10.8, and it is one of the main technical achievements of this paper. It is approximately double the bound obtained by Stolz in [90, Satz 12.7], and we devote § §4-6 and § §9-10 to its proof.Remark 1.6. A key portion of the argument for Theorem 10.8 takes place in Pstrągowski's category of synthetic spectra [77] (cf. [37] for an alternative construction of BP-synthetic spectra). Other users of this category may be interested in our omnibus Theorem 9.19, which relates Adams spectral sequences to synthetic homotopy groups.r. burklund, j. hahn and a. senger for the spectrum Y =C(2)⊗C(η). In more classical language this result is known to experts, and follows from combining Miller's tools with computational results of Davis and Mahowald [28]. In §15, we establish a v 1 -banded vanishing line in the modified HF 2 -Adams spectral sequence for the Moore spectrum C(8) and conclude, in particular, Theorem 1.9.Appendix A. The first part of this appendix is devoted to a technical proof of Theorem 9.19. The theorem provides the means to translate statements about E-based Adams spectral sequences into statements about E-based synthetic spectra, and vice-versa. The proofs in this section are mostly a matter of careful book-keeping.The second part of the appendix contains a computation of the HF 2 -synthetic homotopy groups of the 2-complete sphere through the Toda range. We find that this computation illustrates many of the s...