1 The main point of this talk The main point of this talk The 3-primary Arf-Kervaire invariant problem is still open. We have a program for solving it similar to what we did for p = 2. We are missing a crucial ingredient. Defining the problem The Arf-Kervaire invariant problem for a prime p is to determine the fate of the elements θ j = h 2 j for p = 2 b j−1 for p > 2 ∈ Ext 2,2p j (p−1) A (Z/p,...

where /4(G) denotes the completion of the Burnside ring of G with respect to the ideal of virtual G-sets of degree 0. The conjecture was proved for G = Z/(2) by W. H. Lin [5], [3] and for G = Z/(p), where p is an odd prime, by J. H. C. Gunawardena [4]. In this note we outline a proof for G cyclic. We will assume that G has prime power order as the general case follows easily. The proofs cited a...

Let U denote the category of unstable modules over the mod 2 Steenrod algebra A, and let Ext(−) = ExtsU(−,ΣZ2). Let M∞ = H̃∗(ΣCP∞ + ) denote the unstable A-module with nonzero classes xi such that i is odd and positive, and Sqx2k+1 = ( k j ) x2(j+k)+1. Then H∗(U) ≈ U(M∞), where the left side is the mod 2 cohomology of the infinite unitary group, and the right side the free unstable A-algebra gen...

Let n ≥ 1 and let p be any prime. Also, let En be the LubinTate spectrum, Gn the extended Morava stabilizer group, and K(n) the nth Morava K-theory spectrum. Then work of Devinatz and Hopkins and some results due to Behrens and the first author of this note, show that if X is a finite spectrum, then the localization LK(n)(X) is equivalent to the homotopy fixed point spectrum (LK(n)(En ∧ X))hGn ...

For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture more and more information according to the injective dimension of the images of the functor. The categories are obtained by using truncated versions of resolu...

MU*(S°) (2) Tor*,* \MU*(X)9Z)=*H*(X;Z) by P. E. Conner and L. Smith [9]. Recently N. A. Baas [4] has written an excellent account of the Adams spectral sequence (1), and J. F. Adams has made a thorough analysis of universal coefficient theorems such as (2) in Lecture 1 of [ l ] . In §1 we announce several solutions to the problem—when is MU*(X) isomorphic to the inverse limit of the complex cob...

In this paper we explore the isotropic stable motivic homotopy category constructed from usual category, following work of Vishik on motives (see [28]), by killing anisotropic varieties. particular, focus cohomology operations in realm and study structure Steenrod algebra. Then, construct an version Adams spectral sequence apply it to find a complete description groups sphere spectrum, which ha...

Let A denote the mod 2 Steenrod algebra (see Steenrod and Epstein [28]). The problem of computing its cohomology H∗,∗(A) is of great importance in algebraic topology, for this bigraded commutative algebra is the E2 term of the Adams spectral sequence (see Adams [1]) converging to the stable homotopy groups of spheres. But despite intensive investigation for nearly half a century, the structure ...

This note is intended to draw attention to some phenomena arising naturally within the framework of brave new ring spectra which has recently been constructed by May et al [5]. Actually, our first contact with the algebraic aspects was in the framework of Robinson’s theory of A∞ ring spectra [2, 3], however their most natural interpretation seems to be within the commutative theory. Since writi...