Non-factorisation of Arf-Kervaire classes through ${\mathbb RP}^{\infty} \wedge {\mathbb RP}^{\infty}$

The Arf-Kervaire Invariant of framed manifolds

This work surveys classical and recent advances around the existence of exotic differentiable structures on spheres and its connection to stable homotopy theory.

The Arf-Kervaire invariant of framed manifolds as an obstruction to embeddability

We prove that no $14$-connected (resp. $30$-connected) stably parallelizable manifold $N^{30}$ (resp. $N^{62}$) of dimension $30$ (resp. $62$) with the Arf-Kervaire invariant 1 can be smoothly embedded into $\mathbb{R}^{36}$ (resp. $\mathbb{R}^{83}$).

球面稳定同调群的几何方法 II；阿夫-凯瓦里不变量

Geometric approach to stable homotopy groups of spheres II; Arf-Kervaire Invariants

本文通过几何方法解决球面稳定同调群中的凯瓦里不变量问题，引入了新的对称结构并证明了相关定理。

The non-multiplicativity of the signature modulo 8 of a fibre bundle is an Arf-Kervaire invariant

It was proved by Chern, Hirzebruch and Serre that the signature of a fibre bundle is multiplicative if the fundamental group of the base acts trivially on the cohomology ring of the fibre with real coefficients, in which case the signature of the total space equals the product of the signatures of base and fibre. Hambleton, Korzeniewski and Ranicki proved that in any case the signature is multiplicative modulo 4. In this paper we present two results concerning the multiplicativity modulo 8: firstly we identify the obstruction to multiplicativity modulo 8 with the Arf-Kervaire invariant of a Po…

Kairvaire Problems in Stable Homotopy Theory

The Kervaire Problem is an unsolved problem in Stable Homotopy Theory. The first interesting example is in dimension $30$. There exists a closed stably-parallelizable manifold $\tilde{M}^{30}$ with Arf-Kervaire invariant 1. It is unknown, if such a manifold exists in dimension $126$? The goal of the paper is to recall the construction of the manifold $M^{30}$ and investigate its properties.

Generalised geometric weak conjecture on spherical classes and non-factorisation of Kervaire invariant one elements

This paper is on the Curtis conjecture. We show that the image of the Hurewicz homomorhism $h:π_*Q_0S^0\to H_*(Q_0S^0;\mathbb{Z})$, when restricted to product of positive dimensional elements, is determined by $\mathbb{Z}\{h(η^2),h(ν^2),h(σ^2)\}$. Localised at $p=2$, this proves a geometric version of a result of Hung and Peterson for the Lannes-Zarati homomorphism. We apply this to show that, for $p=2$ and $G=O(1)$ or any prime $p$ and $G$ any compact Lie group with Lie algebra $\mathfrak{g}$ so that $\dim\mathfrak{g}>0$, the composition $${_pπ_*}QΣ^{n\dim\mathfrak{g}}BG_+^{\wedge n}\to {_pπ_…