Let G be a finite group and let F be a family of subgroups of G. We introduce a class of G-equivariant spectra that we call F-nilpotent. This definition fits into the general theory of torsion, ...complete, and nilpotent objects in a symmetric monoidal stable ∞-category, with which we begin. We then develop some of the basic properties of F-nilpotent G-spectra, which are explored further in the sequel to this paper.
In the rest of the paper, we prove several general structure theorems for ∞-categories of module spectra over objects such as equivariant real and complex K-theory and Borel-equivariant MU. Using these structure theorems and a technique with the flag variety dating back to Quillen, we then show that large classes of equivariant cohomology theories for which a type of complex-orientability holds are nilpotent for the family of abelian subgroups. In particular, we prove that equivariant real and complex K-theory, as well as the Borel-equivariant versions of complex-oriented theories, have this property.
We prove that the 2-primary π₆₁ is zero. As a consequence, the Kervaire invariant element θ₅ is contained in the strictly defined 4-fold Toda bracket 〈2, θ₄, θ₄, 2〉. Our result has a geometric ...corollary: the 61-sphere has a unique smooth structure, and it is the last odd dimensional case — the only ones are S¹, S³, S⁵ and S⁶¹. Our proof is a computation of homotopy groups of spheres. A major part of this paper is to prove an Adams differential d₃(D₃) = B₃. We prove this differential by introducing a new technique based on the algebraic and geometric Kahn-Priddy theorems. The success of this technique suggests a theoretical way to prove Adams differentials in the sphere spectrum inductively by use of differentials in truncated projective spectra.
This volume contains the proceedings of the conference Homotopy Theory: Tools and Applications, in honor of Paul Goerss's 60th birthday, held from July 17-21, 2017, at the University of Illinois at ...Urbana-Champaign, Urbana, IL.The articles cover a variety of topics spanning the current research frontier of homotopy theory. This includes articles concerning both computations and the formal theory of chromatic homotopy, different aspects of equivariant homotopy theory and $K$-theory, as well as articles concerned with structured ring spectra, cyclotomic spectra associated to perfectoid fields, and the theory of higher homotopy operations.
Let A denote the Steenrod algebra over the field of characteristic two, F2. Singer's algebraic transfer, introduced by Singer in his work (Singer (1989) 27), is a rather effective tool for unraveling ...the intricate structure of the mod-two cohomology of the Steenrod algebra, ExtA⁎,⁎(F2,F2). In the present study, we aim to investigate the behavior of this algebraic transfer for rank five in the generic family of internal degree n:=ℓ(2s−1)+k⋅2s, wherein ℓ=4,8≤k≤11,k≠9, and s is any positive integer. The principal results obtained lead to interesting conclusions regarding the image of algebraic transfers in ranks 5, 6, 9, and 10. As direct consequences, within the bidegrees (5,5+n), Singer's conjecture on the monomorphism of algebraic transfers remains upheld.
Let PUn denote the projective unitary group of rank n, and let BPUn be its classifying space. We show that the p-primary subgroup of H2p+6(BPUn;Z) is trivial, where p is an odd prime.
We prove a blow-up formula for Dolbeault cohomologies of compact complex manifolds by introducing relative Dolbeault cohomology. As corollaries, we present a uniform proof for bimeromorphic ...invariance of (•,0)- and (0,•)-Hodge numbers on a compact complex manifold, and obtain the equality for the numbers of the blow-ups and blow-downs in the weak factorization of the bimeromorphic map between two compact complex manifolds with equal (1,1)-Hodge number or equivalently second Betti number. Many examples of the latter one are listed. Inspired by these, we obtain the bimeromorphic stability for degeneracy of the Frölicher spectral sequences at E1 on compact complex threefolds and fourfolds.
Nous démontrons une formule d'éclatement pour la cohomologie de Dolbeault des variétés complexes compactes en introduisant la cohomologie relative de Dolbeault. Nous présentons aussi comme corollaire une approche unifiée de l'invariance des nombres de Hodge de type (⁎,0) et (0,⁎) par biméromorphisme. Nous obtenons l'égalité entre le nombre des éclatements et le nombre des contractions d'une factorisation faible d'une application biméromorphe entre deux variétés complexes compactes ayant le même nombre de Hodge de type (1,1) ou, équivalentement, le même second nombre de Betti. Nous donnerons des exemples dans le dernier cas et nous établirons la stabilité biméromorphe de la dégénérescence en E1 de la suite spectrale de Frölicher sur les variétés complexes compactes de dimensions trois et quatre.
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