## Wayne State University Dissertations

#### Access Type

Open Access Dissertation

January 2020

Dissertation

Ph.D.

Mathematics

Daniel Isaksen

#### Abstract

Computing the stable homotopy groups of the sphere spectrum is one of the most important problems of stable homotopy theory. Focusing on the 2-complete stable homotopy groups instead of the integral homotopy groups, the Adams spectral sequence appears to be one of the most effective tools to compute the homotopy groups. The spectral sequence has been studied by J. F. Adams, M. Mahowald, M. Tangora, J. P. May and others.

In 1999, Morel and Voevodsky introduced motivic homotopy theory. One of its consequences is the realization that almost any object studied in classical algebraic topology could be given a motivic analog. In particular, we can define the motivic Steenrod algebra $\mathbf{A}$, the motivic stable homotopy groups of spheres \cite{VM-scheme} and the motivic Adams spectral sequence. In the motivic perspective, there are many more non-zero classes in the motivic Adams spectral sequence, which allows the detection of otherwise elusive phenomena. Also, the additional motivic weight grading can eliminate possibilities which appear plausible in the classical perspective.

To run the motivic Adams spectral sequence, one begins with Ext$_{\mathbf{A}}(\mathbb{M}_2,\mathbb{M}_2)$. The algebra Ext$_{\mathbf{A}}(\mathbb{M}_2,\mathbb{M}_2)$ is infinitely generated and irregular. A natural approach is to look for systematic phenomena in Ext$_{\mathbf{A}}(\mathbb{M}_2,\mathbb{M}_2)$. One potential candidate is the wedge family in Ext$_{\mathbf{A}}(\mathbb{M}_2,\mathbb{M}_2)$.

The classical wedge family was studied by M. Mahowald and M. Tangora \cite{Maho-Tango-wedge}. It is a subset of the cohomology Ext$_{\mathbf{A}_{\mathrm{cl}}}(\mathbb{F}_2,\mathbb{F}_2)$ of the classical Steenrod algebra, consisting of non-zero elements $P^ig^j \lambda$ and $g^j t$ in which $\lambda$ is in $\boldsymbol\Lambda$, $t$ is in $\mathbf{T}$, $i\ge 0$ and $j\ge 0$. The sets $\boldsymbol\Lambda$ and $\mathbf{T}$ are specific subsets of Ext$_{\mathbf{A}_{\mathrm{cl}}}(\mathbb{F}_2,\mathbb{F}_2)$. The wedge family gives an infinite wedge-shaped diagram inside the cohomology of the classical Steenrod algebra, which fills out an angle with vertex at $g^2$ in degree (40,8) (i.e. $g^2$ has stem 40 and Adams filtration 8), bounded above by the line $f=\frac{1}{2}s-12$, parallel to the Adams edge \cite{Adams-ped}, and bounded below by the line $s=5f$, in which $f$ is the Adams filtration and $s$ is the stem. The wedge family is a large piece of Ext$_{\mathbf{A}_{\mathrm{cl}}}(\mathbb{F}_2,\mathbb{F}_2)$ which is regular, of considerable size and easy to understand.

Using this idea we build the motivic version of the wedge. However, it appears to be more complicated than the classical one. The motivic wedge family takes the same position and same shape as the classical one. However the vertex of the motivic wedge is at $\tau g^2$ in degree $(40,8,23)$ having weight 23. Note that $g^2$ in degree $(40,8,24)$ does not survive the motivic May spectral sequence \cite{I-stem}. Our main result, Theorem \ref{def-wedge}, states that the subsets $\tau^k \mathbf{P}^i\mathbf{g}^j\lambda$ are non-empty and consist of non-zero elements for all $\lambda$ in $\boldsymbol\Lambda$.

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