On the cooperation algebra of the connective Adams summand

Uniqueness of $E_\infty$ structures for connective covers

We refine our earlier work on the existence and uniqueness of E-infinity structures on K-theoretic spectra to show that at each prime p, the connective Adams summand has an essentially unique structure as a commutative S-algebra. For the p-completion we show that the McClure-Staffeldt model for it is equivalent as an E-infinity ring spectrum to the connective cover of the periodic Adams summand. We establish Bousfield equivalence between the connective cover, c(E_n), of the Lubin-Tate spectrum E_n and BP<n> and propose c(E_n) as an E-infinity approximation to the latter.

Upper triangular matrices and operations in odd primary connective K-theory

We prove analogues for odd primes of results of Snaith and Barker-Snaith. Let l denote the p-complete connective Adams summand and consider the group of left l-module automorphisms of l smash l in the stable homotopy category which induce the identity on mod p homology. We prove a group isomorphism between this group and a certain group of infinite invertible upper triangular matrices with entries in the p-adic integers. We determine information about the matrix corresponding to the automorphism 1 smash Psi^q, where Psi^q is the Adams operation and q is an integer which generates the p-adic un…

The homotopy groups of the spectrum Tmf

We use the structure of the homotopy groups of the connective spectrum tmf of topological modular forms and the elliptic and Adams-Novikov spectral sequences to compute the homotopy groups of the non-connective version, Tmf, of that spectrum. This is done separately for the localizations at 2, 3 and higher primes.

Eilenberg-MacLane空间K(Z_2,2)的连通KO理论，I：E_2页

The connective KO-theory of the Eilenberg-MacLane space K(Z_2,2), I: the E_2 page

本研究计算了收敛于第二模2 Eilenberg-MacLane空间连通KO理论的Adams谱序列E_2页，分析了对应上同调作为Steenrod代数子代数模的结构，完整谱序列分析将后续发表。

Logarithmic structures on topological K-theory spectra

We study a modified version of Rognes' logarithmic structures on structured ring spectra. In our setup, we obtain canonical logarithmic structures on connective K-theory spectra which approximate the respective periodic spectra. The inclusion of the p-complete Adams summand into the p-complete connective complex K-theory spectrum is compatible with these logarithmic structures. The vanishing of appropriate logarithmic topological Andre-Quillen homology groups confirms that the inclusion of the Adams summand should be viewed as a tamely ramified extension of ring spectra.

$Γ$-cohomology of rings of numerical polynomials and $E_\infty$ structures on K-theory

We investigate Gamma-cohomology of some commutative cooperation algebras E_*E associated with certain periodic cohomology theories. For KU and E(1), the Adams summand at a prime p, and for KO we show that Gamma-cohomology vanishes above degree 1. As these cohomology groups are the obstruction groups in the obstruction theory developed by Alan Robinson we deduce that these spectra admit unique E infinity structures. As a consequence we obtain an E infinity structure for the connective Adams summand. For the Johnson-Wilson spectrum E(n) with n > 0 we establish the existence of a unique E infinit…