Root invariants in the Adams spectral sequence

Some root invariants at the prime 2

The first part of this paper consists of lecture notes which summarize the machinery of filtered root invariants. A conceptual notion of "homotopy Greek letter element" is also introduced, and evidence is presented that it may be related to the root invariant. In the second part we compute some low dimensional root invariants of v_1-periodic elements at the prime 2.

Beta-elements and divided congruences

The f-invariant is an injective homomorphism from the 2-line of the Adams-Novikov spectral sequence to a group which is closely related to divided congruences of elliptic modular forms. We compute the f-invariant for two infinite families of beta-elements and explain the relation of the arithmetic of divided congruences with the Kervaire invariant one problem.

C₂等变Adams谱序列中的代数红移

Algebraic redshift in the $C_2$-equivariant Adams spectral sequence

研究C₂等变稳定同调中v_n周期现象，通过C₂等变Adams谱序列在素数2处的视角，探讨相关类在某些有限生成子代数的上同调中的构造与检测，定义v_n周期或v_n torsion类，并展示染色过滤。

R-motivic stable stems

We compute some R-motivic stable homotopy groups. For $s - w \leq 11$, we describe the motivic stable homotopy groups $π_{s,w}$ of a completion of the R-motivic sphere spectrum. We apply the $ρ$-Bockstein spectral sequence to obtain R-motivic Ext groups from the C-motivic Ext groups, which are well-understood in a large range. These Ext groups are the input to the R-motivic Adams spectral sequence. We fully analyze the Adams differentials in a range, and we also analyze hidden extensions by $ρ$, 2, and $η$. As a consequence of our computations, we recover Mahowald invariants of many low-dimens…

Motivic invariants of p-adic fields

We provide a complete analysis of the motivic Adams spectral sequences converging to the bigraded coefficients of the 2-complete algebraic Johnson-Wilson spectra BPGL<n> over p-adic fields. These spectra interpolate between integral motivic cohomology (n=0), a connective version of algebraic K-theory (n=1), and the algebraic Brown-Peterson spectrum. We deduce that, over p-adic fields, the 2-complete BPGL<n> split over 2-complete BPGL<0>, implying that the slice spectral sequence for BPGL collapses. This is the first in a series of two papers investigating motivic invariants of p-adic fields, a…

The Bredon-Landweber region in $C_2$-equivariant stable homotopy groups

We use the $C_2$-equivariant Adams spectral sequence to compute part of the $C_2$-equivariant stable homotopy groups $π^{C_2}_{n,n}$. This allows us to recover results of Bredon and Landweber on the image of the geometric fixed-points map from the equivariant homotopy group $π^{C_2}_{n,n}$ to the classical $π_0$. We also recover results of Mahowald and Ravenel on the Mahowald root invariants of the elements $2^k$.

Invariants of links with flat connections in their complements.II. Holonomy R-matrices related to quantized universal enveloping algebras at roots of 1

Holonomy R-matrices parametrized by finite-dimensional representations are constructed for quantized universal enveloping algebras of simple Lie algebras at roots of 1.

Modular invariants detecting the cohomology of BF_4 at the prime 3

Attributed to J F Adams is the conjecture that, at odd primes, the mod-p cohomology ring of the classifying space of a connected compact Lie group is detected by its elementary abelian p-subgroups. In this note we rely on Toda's calculation of H^*(BF_4;F_3) in order to show that the conjecture holds in case of the exceptional Lie group F_4. To this aim we use invariant theory in order to identify parts of H^*(BF_4;F_3) with invariant subrings in the cohomology of elementary abelian 3-subgroups of F_4. These subgroups themselves are identified via the Steenrod algebra action on H^*(BF_4;F_3).

Z/4×Z/4 群的 Gromov-Lawson-Rosenberg 猜想

The Gromov-Lawson-Rosenberg Conjecture for Z/4xZ/4

本研究通过 Adams 谱序列计算分类空间连通实 k-同调，结合两类α不变量核检测定理，证明了Z/4×Z/4群的Gromov-Lawson-Rosenberg猜想，同时确定了相关谱序列微分并研究了对应子Hopf代数的谱序列上积结构。

On the homotopy groups of E(n)-local spectra with unusual invariant ideals

Let E(n) and T(m) for nonnegative integers n and m denote the Johnson-Wilson and the Ravenel spectra, respectively. Given a spectrum whose E(n)_*-homology is E(n)_*(T(m))/(v_1,...,v_{n-1}), then each homotopy group of it estimates the order of each homotopy group of L_nT(m). We here study the E(n)-based Adams E_2-term of it and present that the determination of the E_2-term is unexpectedly complex for odd prime case. At the prime two, we determine the E_{infty}-term for pi_*(L_2T(1)/(v_1)), whose computation is easier than that of pi_*(L_2T(1)) as we expect.