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Register a new userThe triviality of the 61-stem in the stable homotopy groups of spheres
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We prove that the 2-primary $pi_{61}$ is zero. As a consequence, the Kervaire invariant element $theta_5$ is contained in the strictly defined 4-fold Toda bracket $langle 2, theta_4, theta_4, 2rangle$. 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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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.
Let $S(V)$ be a complex linear sphere of a finite group $G$. %the space of unit vectors in a complex representation $V$ of a finite group $G$. Let $S(V)^{*n}$ denote the $n$-fold join of $S(V)$ with itself and let $aut_G(S(V)^*)$ denote the space of $G$-equivariant self homotopy equivalences of $S(V)^{*n}$. We show that for any $k geq 1$ there exists $M>0$ which depends only on $V$ such that $|pi_k aut_G(S(V)^{*n})| leq M$ is for all $n gg 0$.
Let $G$ be a finite group. Let $U_1,U_2,dots$ be a sequence of orthogonal representations in which any irreducible representation of $oplus_{n geq 1} U_n$ has infinite multiplicity. Let $V_n=oplus_{i=1}^n U_n$ and $S(V_n)$ denote the linear sphere of unit vectors. Then for any $i geq 0$ the sequence of group $dots rightarrow pi_i operatorname{map}^G(S(V_n),S(V_n)) rightarrow pi_i operatorname{map}^G(S(V_{n+1}),S(V_{n+1})) rightarrow dots$ stabilizes with the stable group $oplus_H omega_i(BW_GH)$ where $H$ runs through representatives of the conjugacy classes of all the isotropy group of the points of $S(oplus_n U_n)$.
These are notes for a five lecture series intended to uncover large-scale phenomena in the homotopy groups of spheres using the Adams-Novikov Spectral Sequence. The lectures were given in Strasbourg, May 7-11, 2007.
We survey computations of stable motivic homotopy groups over various fields. The main tools are the motivic Adams spectral sequence, the motivic Adams-Novikov spectral sequence, and the effective slice spectral sequence. We state some projects for future study.
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