The Hopf invariant one problem

Invariant elements in the dual Steenrod algebra

‎In this paper‎, ‎we investigate the invariant elements of the dual mod $p$ Steenrod subalgebra ${mathcal{A}_p}^*$ under the conjugation map $chi$ and give bounds on the dimensions of $(chi-1)({mathcal{A}_p}^*)_d$‎, ‎where $({mathcal{A}_p}^*)_d$ is the dimension of ${mathcal{A}_p}^*$ in degree $d$‎.

Stable Homotopy Groups of Spheres and The Hopf Invariant One Problem

Involutory Hopf Group-coalgebras and Flat Bundles over 3-manifolds

Given a group π, we use involutary Hopf π-coalgebras to define a scalar invariant of flat π-bundles over 3-manifolds. When π = 1, this invariant equals to the one of 3-manifolds constructed by Kuperberg from involutary Hopf algebras. We give examples which show that this invariant is not trivial.

if-THEORY AND THE HOPF INVARIANT

[Received 29 December 1964] Introduction THE non-existence of elements of Hopf invariant one in ir%n-^(S ), for n ^ 1, 2, 4, or 8, was established in (1) by the use of secondary cohomology operations. The main purpose of this paper is to show how the use of primary operations in .K"-theory provides an extremely simple alternative proof of this result. In fact ^-theory proofs have already been g...

HOPF BIFURCATION CONTROL WITH PD CONTROLLER

In this paper, we investigate the problem of bifurcation control for a delayed logistic growth model. By choosing the timedelay as the bifurcation parameter, we present a Proportional - Derivative (PD) Controller to control Hopf bifurcation. We show that the onset of Hopf bifurcation can be delayed or advanced via a PD Controller by setting proper controlling parameter. Under consideration mode...