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

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.

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

As an application of the upper triangular technology method of (V.P. Snaith: {\em Stable homotopy -- around the Arf-Kervaire invariant}; Birkhäuser Progress on Math. Series vol. 273 (April 2009)) it is shown that there do not exist stable homotopy classes of $ {\mathbb RP}^{\infty} \wedge {\mathbb RP}^{\infty}$ in dimension $2^{s+1}-2$ with $s \geq 2$ whose composition with the Hopf map to $ {\mathbb RP}^{\infty}$ followed by the Kahn-Priddy map gives an element in the stable homotopy of spheres of Arf-Kervaire invariant one.

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…

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。

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.

Browder-Livesay filtrations and the example of Cappell and Shaneson

Let $M^3$ be a 3-dimensional manifold with fundamental group $π_1(M)$ which contains a quaternion subgroup $Q$ of order 8. In 1979 Cappell and Shaneson constructed a nontrivial normal map $ f\colon M^3\times T^2\to M^3\times S^2$ which cannot be detected by simply connected surgery obstructions along submanifolds of codimension 0, 1, or 2, but it can be detected by the codimension 3 Kervaire-Arf invariant. The proof of non-triviality of $σ(f)\in L_5(π_1(M))$ is based on consideration of a Browder-Livesay filtration of a manifold $X$ with $π_1(X)\cong π_1(M)$. For a Browder-Livesay pair $Y^{n-1…