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On the nonexistence of elements of Kervaire invariant one. (English) Zbl 1366.55007
This is a landmark paper in algebraic topology solving the Kervaire invariant one problem except for in dimension \(126\). This problem arose in work of Kervaire and Milnor. In particular, they wanted to compute the set of diffeomorphism classes of manifolds homeomorphic to spheres and compare it to cobordism of framed manifolds. One version of the Kervaire invariant one problem is: In which dimensions \(n\) is every (stably) framed manifold framed bordant to a manifold homeomorphic to a sphere? Framed manifolds not framed bordant to a sphere are called of Kervaire invariant one. In this formulation, the Kervaire problem has concrete implications for the study of exotic spheres.
Via the Pontryagin-Thom construction, framed bordism corresponds to the stable homotopy groups of spheres and Browder managed to reformulate the Kervaire problem completely in this language: Manifolds of Kervaire invariant one can exist only in dimensions of the form \(n=2^{j+1}-2\) and they exist here if and only if \(h_j^2\) is a permanent cycle in the Adams spectral sequence for the sphere at the prime \(2\). In this formulation, the problem is a higher version of the Hopf invariant one problem, which asks in the stable homotopy formulation for which \(j\) the class \(h_j\) is a permanent cycle; this was solved by Adams in 1960.
In dimensions \(n= 2,6\) and \(14\), Kervaire invariant one manifolds are easily constructed geometrically as \(\mathbb{KP}^2\times \mathbb{KP}^2\) for \(\mathbb{K} = \mathbb{C},\mathbb{H}, \mathbb{O}\). In dimensions \(30\) and \(62\) the existence of Kervaire invariant one manifolds was shown by stable homotopy techniques [Z. Xu, Geom. Topol. 20, No. 3, 1611–1624 (2016; Zbl 1352.55007)]. The paper under review shows that there is no Kervaire invariant one manifold in dimension bigger than \(126\) (with the case \(n=126\) remaining open).
To prove their main theorem, the authors construct a ring spectrum \(\Omega\) with the following properties: It is \(256\)-periodic, \(\pi_{-2}\Omega =0\) and every class \(\Theta_j\in \pi_{2^{j+1}-1}\mathbb{S}\) representing \(h_j^2\) maps non-trivially to \(\pi_*\Omega\) (this is called the detection theorem). These properties directly imply the theorem.
To construct \(\Omega\), they start with the real bordism spectrum \(MU_\mathbb{R}\); this is a \(C_2\)-spectrum that is essentially the complex bordism spectrum \(MU\) with its complex conjugation action. The word “essentially” hides several subtleties, having to do with working in a genuine equivariant category of spectra. The foundations were build by Peter May and various coworkers over the last decades; several refinements were necessary for the present work, resulting in two lenghty appendices that comprise almost half of the paper.
One of these refinements is the norm construction. This way, we can multiplicatively norm up \(MU_\mathbb{R}\) to a \(C_8\)-spectrum \(N_{C_2}^{C_8}MU_\mathbb{R}\) (that is on underlying spectra equivalent to \(MU^{\wedge 4}\)). Inverting a carefully chosen element \(D\) and taking \(C_8\)-homotopy fixed points produces the spectrum \(\Omega\).
To show that \(\pi_{-2}\Omega = 0\), the authors use another innovation in equivariant homotopy theory: the slice spectral sequence. This is a spectral sequence (inspired by motivic homotopy theory) that converges to the homotopy groups of fixed points of a given equivariant spectrum and is based on the slice tower. The latter is a variant of the Postnikov tower based on the use of representation spheres. One of the most difficult technical theorems in the paper is to show that the layers of this tower (called slices) for \(N_{C_2}^{C_8}MU_\mathbb{R}\) are of the form \(H\underline{\mathbb{Z}} \wedge W\), where \(W\) is induced up from a regular representation sphere of a subgroup \(\{e\}\neq H \subset C_8\). The homotopy groups of the slices are easily computed in terms of Bredon homology of representation spheres and this gives a computation of the \(E_2\)-term of the slice spectral sequence. In particular, one can show that the column contributing to \(\pi_{-2}\) of the \(C_8\)-fixed points of \(N_{C_2}^{C_8} MU_\mathbb{R}\) just vanishes in the slice spectral sequence (and similarly after suspension by a regular representation sphere); this implies that also the fixed points of \(D^{-1}N_{C_2}^{C_8} MU_\mathbb{R}\) have vanishing \(\pi_{-2}\). An additional argument identifies these fixed points with the homotopy fixed points \(\Omega\).
While deducing some periodicity for \(\Omega\) would be easy using versions of the Devinatz-Hopkins-Smith nilpotence theorem, the specific periodicity for \(\Omega\) is more subtle to prove. The \(E_2\)-term of its homotopy fixed point spectral sequence (HFPSS) is periodic and using computations in the slice spectral sequence the authors can show that a suitable power of the periodicity generator is a permanent cycle.
The last missing step is the detection theorem. For this, the authors use computations from [K. Shimomura, Hiroshima Math. J. 11, 499–513 (1981; Zbl 0485.55013)] to see which elements \(x\) in the Adams-Novikov spectral sequence (ANSS) for the sphere could map to \(h_j^2\). There is an algebraically defined map from this ANSS to the HFPSS for \(\Omega\) converging to the map \(\pi_*\mathbb{S} \to \pi_*\Omega\) and one has to show that these classes \(x\) are mapped to something non-trivial. The \(E_2\)-term of the HFPSS of \(\Omega\) is mapped via Lubin-Tate theory down to something much smaller, where the classes \(x\) are still detected and this finishes the proof. Note that the detection theorem depends on a very careful choice of the periodicity class \(D\).
Since the preprint status of this paper, its innovations in equivariant homotopy theory have found several applications. For example, the norm construction has been applied to the study of topological cyclic homology and the understanding of commutativity of equivariant ring spectra; see e.g. [A. J. Blumberg and M. A. Hill, Adv. Math. 285, 658–708 (2015; Zbl 1329.55012)]. The slice tower has been applied in the study of Anderson duality, e.g. in [N. Ricka, Glasg. Math. J. 58, No. 3, 649–676 (2016; Zbl 1350.55012)], and in a cellular construction of \(BP\mathbb{R}\) in recent work by D. Wilson.

MSC:
55P91 Equivariant homotopy theory in algebraic topology
55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
55T99 Spectral sequences in algebraic topology
57R60 Homotopy spheres, Poincaré conjecture
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
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