arXiv subjects

Zhouli Xu

Publications and source records attributed to Zhouli Xu.

27 records · Page 2Linked to original sources

arXiv2020

Stable homotopy groups of spheres

We discuss the current state of knowledge of stable homotopy groups of spheres. We describe a new computational method that yields a streamlined computation of the first 61 stable homotopy groups, and gives new information about the stable homotopy groups in dimensions 62 through 90. The method relies more heavily on machine computations than previous methods, and is therefore less prone to error. The main mathematical tool is the Adams spectral sequence.

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arXiv2019

On the E2-term of the bo-Adams spectral sequence

The E_1-term of the (2-local) bo-based Adams spectral sequence for the sphere spectrum decomposes into a direct sum of a v_1-periodic part, and a v_1-torsion part. Lellmann and Mahowald completely computed the d_1-differential on the v_1-periodic part, and the corresponding contribution to the E_2-term. The v_1-torsion part is harder to handle, but with the aid of a computer it was computed through the 20-stem by Davis. Such computer computations are limited by the exponential growth of v_1-torsion in the E_1-term. In this paper, we introduce a new method for computing the contribution of the v_1-torsion part to the E_2-term, whose input is the cohomology of the Steenrod algebra. We demonstrate the efficacy of our technique by computing the bo-Adams spectral sequence beyond the 40-stem.

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arXiv2019

Intersection Forms of Spin 4-Manifolds and the Pin(2)-Equivariant Mahowald Invariant

In studying the "11/8-Conjecture" on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of Pin(2)-equivariant stable maps between certain representation spheres. In this paper, we present a complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald invariants. As a geometric application of our result, we prove a "10/8+4"-Theorem. We prove our theorem by analyzing maps between certain finite spectra arising from BPin(2) and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known results on the stable homotopy groups of spheres, and the $j$-based Atiyah-Hirzebruch spectral sequence.

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arXiv2018

Some extensions in the Adams spectral sequence and the 51-stem

We show a few nontrivial extensions in the classical Adams spectral sequence. In particular, we compute that the 2-primary part of $π_{51}$ is $\mathbb{Z}/8\oplus\mathbb{Z}/8\oplus\mathbb{Z}/2$. This was the last unsolved 2-extension problem left by the recent works of Isaksen and the authors (\cite{Isa1}, \cite{IX}, \cite{WX1}) through the 61-stem. The proof of this result uses the $RP^\infty$ technique, which was introduced by the authors in \cite{WX1} to prove $π_{61}=0$. This paper advertises this method through examples that have simpler proofs than in \cite{WX1}.

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arXiv2018

Hurewicz Images of Real Bordism Theory and Real Johnson-Wilson Theories

We show that the Hopf elements, the Kervaire classes, and the $\barκ$-family in the stable homotopy groups of spheres are detected by the Hurewicz map from the sphere spectrum to the $C_2$-fixed points of the Real Brown-Peterson spectrum. A subset of these families is detected by the $C_2$-fixed points of Real Johnson-Wilson theory $E\mathbb{R}(n)$, depending on $n$.

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arXiv2017

The triviality of the 61-stem in the stable homotopy groups of spheres

We prove that the 2-primary $π_{61}$ is zero. As a consequence, the Kervaire invariant element $θ_5$ is contained in the strictly defined 4-fold Toda bracket $\langle 2, θ_4, θ_4, 2\rangle$. 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 $S^1, S^3, S^5$ and $S^{61}$. Our proof is a computation of homotopy groups of spheres. A major part of this paper is to prove an Adams differential $d_3(D_3) = B_3$. 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.

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arXiv2016

The algebraic Atiyah-Hirzebruch spectral sequence of real projective spectra

In this note, we use Curtis's algorithm and the Lambda algebra to compute the algebraic Atiyah-Hirzebruch spectral sequence of the suspension spectrum of $\mathbb{R}P^\infty$ with the aid of a computer, which gives us its Adams $E_2$-page in the range of $t<72$. We also compute the transfer map on the Adams $E_2$-pages. These data are used in our computations of the stable homotopy groups of $\mathbb{R}P^\infty$ in [6] and of the stable homotopy groups of spheres in [7].

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arXiv2014

Motivic Stable Homotopy and the Stable 51 and 52 Stems

We establish a differential $d_2(D_1)=h_0^2h_3g_2$ in the $51$-stem of the Adams spectral sequence at the prime $2$, which gives the first correct calculation of the stable 51 and 52 stems. This differential is remarkable since we know of no way to prove it without recourse to the motivic Adams spectral sequence. It is the last undetermined differential in the range of the first author's detailed calculations of the $n$-stems for $n<60$ [6]. This note advertises the use of the motivic Adams spectral sequence to obtain information about classical stable homotopy groups.

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arXiv2014

The strong Kervaire invariant problem in dimension 62

Using a Toda bracket computation $\langle θ_4, 2, σ^2\rangle$ due to Daniel C. Isaksen [11], we investigate the $45$-stem more thoroughly. We prove that $θ_4^2=0$ using a $4$-fold Toda bracket. By [2], this implies that $θ_5$ exists and there exists a $θ_5$ such that $2θ_5=0$. Based on $θ_4^2=0$, we simplify significantly the $9$-cell complex construction in [1] to a $4$-cell complex, which leads to another proof that $θ_5$ exists.

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