Homotopy Groups of Spheres: a Very Basic Introduction

A new family in the stable homotopy groups of spheres

Let $p$ be a prime number greater than three. In this paper, we prove the existence of a new family of homotopy elements in the stable homotopy groups of spheres $pi_{ast}(S)$ which is represented by $h_nh_mtilde{beta}_{s+2}in {rm Ext}_A^{s+4, q[p^n+p^m+(s+2)p+(s+1)]+s}(mathbb{Z}_p,mathbb{Z}_p)$ up to nonzero scalar in the Adams spectral sequence, where $ngeq m+2>5$, $0leq sExt}_A^{s+2,q[(s+2)p...

Detection of a nontrivial element in the stable homotopy groups of spheres

‎Let $p$ be a prime with $pgeq 7$ and $q=2(p-1)$‎. ‎In this paper‎ ‎we prove the existence of a nontrivial product of‎ ‎filtration $s+4$ in the stable homotopy groups of spheres‎. ‎This nontrivial‎ ‎product is shown to be represented up to a nonzero scalar by‎ ‎the product element $widetilde{gamma}_{s}b_{n-1}g_{0}in‎ ‎{Ext}_{mathcal{A}}^{s+4,(p^n+sp^2+sp+s)q+s-3}(mathbb{Z}/p,mathbb{Z}/p)$‎ ‎in ...

On the homotopy groups of spheres in homotopy type theory

The goal of this thesis is to prove that π4(S) ' Z/2Z in homotopy type theory. In particular it is a constructive and purely homotopy-theoretic proof. We first recall the basic concepts of homotopy type theory, and we prove some well-known results about the homotopy groups of spheres: the computation of the homotopy groups of the circle, the triviality of those of the form πk(S) with k < n, and...

Higher Inductive Types in Homotopy Type Theory

Homotopy Type Theory (HoTT) refers to the homotopical interpretation [1] of Martin-Löf’s intensional, constructive type theory (MLTT) [5], together with several new principles motivated by that interpretation. Voevodsky’s Univalent Foundations program [6] is a conception for a new foundation for mathematics, based on HoTT and implemented in a proof assistant like Coq [2]. Among the new principl...

An Introduction to the Homotopy Groups of Spheres