Robert R. Bruner

On the behavior of the algebraic transfer

Let Trk : F 2 ⊗ PH i (BV k ) → Ext k,k+i A (F 2 ,F 2 ) be the alge-GL k braic transfer, which is defined by W. Singer as an algebraic version of the geometrical transfer tr k : π S * ((BV k ) + ) → π S * (S 0 ). It has been shown that the algebraic transfer is highly nontrivial and, more precisely, that Tr k is an isomorphism for k = 1,2,3. However, Singer showed that Tr 5 is not an epimorphism. In this paper, we prove that Tr 4 does not detect the nonzero element g s ∈ Ext 4,12.2s A (F 2 ,F 2 ) for every s > 1. As a consequence, the localized (Sq 0 ) -1 Tr 4 given by inverting the squaring operation Sq° is not an epimorphism. This gives a negative answer to a prediction by Minami.

On Recursive Solutions of a Unit Fraction Equation

We analyze in homological terms the homotopy fixed point spec- trum of a T-equivariant commutative S-algebra R. There is a homological homotopy fixed point spectral sequence with E 2 = H s gp (T; Ht(R; Fp)), converging conditionally to the continuous homology H c s+t (R hT ; Fp) of the homotopy fixed point spectrum. We show that there are Dyer-Lashof op- erations β ǫ Q i acting on this algebra spectral sequence, and that its dif- ferentials are completely determined by those originating on the vertical axis. More surprisingly, we show that for each class x in the E 2r -term of the spectral sequence there are 2r other classes in the E 2r -term (obtained mostly by Dyer-Lashof operations on x) that are infinite cycles, i.e., survive to the E 1 -term. We apply this to completely determine the differentials in the homological homotopy fixed point spectral sequences for the topological Hochschild homology spectra R = THH(B) of many S-algebras, including B = MU, BP , ku, ko and tmf. Similar results apply for all finite sub- groups C ⊂ T, and for the Tate- and homotopy orbit spectral sequences. This work is part of a homological approach to calculating topological cyclic homology and algebraic K-theory of commutative S-algebras. AMS Classification 19D55, 55S12, 55T05 ; 55P43, 55P91

REAL CONNECTIVE K-THEORY AND THE QUATERNION GROUP

We consider the Egyptian fraction equation and discuss techniques for generating solutions. By examining a quadratic recurrence relation modulo a family of primes we have found some 500 new infinite sequences of solutions. We also initiate an investigation of the randomness of the distribution of solutions, and show that there are infinitely many solutions not generated by the aforementioned technique.

The Breden–Löffler Conjecture

Let ko be the real connective K-theory spectrum. We compute ko*BG and ko*BG for groups G whose Sylow 2-subgroup is quaternion of order 8. Using this we compute the coefficients t(ko)G of the G fixed points of the Tate spectrum t(ko) for G = S12 (3) and G = Q8. The results provide a counterexample to the optimistic conjecture of Greenlees and May [9, Conj. 13.4], by showing, in particular, that t(ko)G is not a wedge of Eilenberg-Mac Lane spectra, as occurs for groups of prime order.

An example in the cohomology of augmented algebras

We give a brief exposition of results of Bredon and others on passage to fixed points from stable C 2 equivariant homotopy (where C 2 is the group of order two) and its relation to Mahowalds root invariant. In particular we give Bredons easy equivariant proof that the root invariant doubles the stem; the conjecture of the title is equivalent to the Mahowald-Ravenel conjecture that the root invariant never more than triples the stem. Our main result is to verify by computation that the algebraic analogue of this holds in an extensive range: this improves on results of [Mahowald and Shick 1983].

Ossa's theorem and adams covers

We show that the cohomology of augmented algebras is quite sensitive to changes in the augmentation by exhibiting algebras A, and A,, isomorphic as algebras but not as augmented algebras, such that H*A, is commutative but H*& is not. We shall work exclusively over the field Z, of 2 elements, though there are odd primary analogs of A, and A, whose cohomologies are similarly related. We will need Priddy’s results on the cohomology of Koszul algebras [3], so we begin by summarizing these. Let T{x,} be the tensor algebra generated by the set {xi}, with augmentation E(Xi) = 0. An augmented algebra A is a pre-Koszul algebra if there is an epimorphism of augmented algebras (Y : T{xi} -tA whose kernel is the two-sided ideal generated by elements of the form Cgixj+ CAjXiXj, where g, and fij are in Z,. Clearly we may assume the ai = a(~,) are linearly independent, in which case we call (Y a Koszul presentation and {ai} a set of Koszul generators. The pre-Koszul algebra A is homogeneous if all the gj can be taken to be 0. A homogeneous Koszul algebra is a homogeneous pre-Koszul algebra A such that H*A is generated as an algebra by the cocycles dual to the a;. In practice, this is verified by showing that A has a basis of monomials in the ai of particularly nice form [3, Theorem 5.31. The algebra r{x,l is filtered by letting Fpr{xi) be spanned by all monomials of length p or less. If a is a Koszul presentation of the pre-Koszul algebra A, let &,A = a(FPT{xi}). The associated graded algebra E”A is a homogeneous pre-Koszul algebra with Koszul generators bi, the images of the ai, and relations C J;ib;bj. Note that A is homogeneous iff A = E’A. A Koszul algebra is a pre-Koszul algebra A such that E”A is a homogeneous Koszul algebra. To state the main theorem of [3], let A be a Koszul algebra with Koszul generators (ai ) iczJ}. Let B b e a Z2 basis of A containing 1, the ai, and certain monomials aj,aiz.. . ai n Let SC Unzl . J” be such that for each a E B{ l> there is a unique (il,i2,..., i,,) E S with a = ai,ai2.. . ag. The relations for A may be written

Idempotents, Localizations and Picard groups of A(1)-modules

We show that Ossa’s theorem splitting ku∧BV for elementary abelian groups V follows from general facts about ku∧BZ/2 and Adams covers. For completeness, we also provide the analogous results for ko ∧BV .

A counterexample for lightning flash modules over E(e 1, e 2)

The goal of this article is to make explicit a structured complex whose homology computes the cohomology of the p-profinite completion of the n-fold loop space of a sphere of dimension d=n-m<n. This complex is defined purely algebraically, in terms of characteristic structures of E_n-operads. Our construction involves: the free complete algebra in one variable associated to any E_n-operad; and an element in this free complete algebra, which is associated to a morphism from the operad of L-infinity algebras to an operadic suspension of our E_n-operad. We deduce our main theorem from: a connection between the cohomology of iterated loop spaces and the cohomology of algebras over E_n-operads; and a Koszul duality result for E_n-operads.We analyze the stable isomorphism type of polynomial rings on degree 1 generators as modules over the sub-algebra A(1) = of the mod 2 Steenrod algebra. Since their augmentation ideals are Q_1-local, we do this by studying the Q_i-local subcategories and the associated Margolis localizations. The periodicity exhibited by such modules reduces the calculation to one that is finite. We show that these are the only localizations which preserve tensor products, by first computing the Picard groups of these subcategories and using them to determine all idempotents in the stable category of bounded-below A(1)-modules. We show that the Picard groups of the whole category are detected in the local Picard groups, and show that every bounded-below A(1) -module is uniquely expressible as an extension of a Q_0-local module by a Q_1-local module, up to stable equivalence. Applications include correct, complete proofs of Ossas theorem, applications to Powells work describing connective K-theory of classifying spaces of elementary abelian groups in functorial terms, and Aults work on the hit problem.