Hopf construction map in higher dimensions

范畴Hopf映射

Categorical Hopf map

引入范畴Hopf映射作为以Nora Ganter的范畴圆为纤维的二维球面上的范畴主丛，探讨其与Hopf映射的联系，并给出通过Hopf映射和三维球面上的基本丛gerbe的分解。

A simplicial model for the Hopf map

We give an explicit simplicial model for the Hopf map S^3 -> S^2. For this purpose, we construct a model of S^3 as a principal twisted cartesian product K x_{eta} S^2, where K is a simplicial model for S^1 acting by left multiplication on itself, S^2 is given the simplest simplicial model and the twisting map is eta:(S^2)_n -> (K)_{n-1}. We construct a Kan complex for the simplicial model K of S^1. The simplicial model for the Hopf map is then the projection K x_{eta} S^2 -> S^2.

Simplicial James-Hopf map and decompositions of the unstable Adams spectral sequence for suspensions

We use combinatorial group theory methods to extend the definition of a classical James-Hopf invariant to a simplicial group setting. This allow us to realize certain coalgebra idempotents at sSet -level and obtain a functorial decomposition of the spectral sequence, associated with the lower p-central series filtration on the free simplicial group.

Hopf invariants, topological complexity, and LS-category of the cofiber of the diagonal map for two-cell complexes

Let $X$ be a two-cell complex with attaching map $α\colon S^q\to S^p$, and let $C_X$ be the cofiber of the diagonal inclusion $X\to X\times X$. It is shown that the topological complexity (${\rm TC}$) of $X$ agrees with the Lusternik-Schnirelmann category (${\rm cat}$) of $C_X$ in the (almost stable) range $q\leq2p-1$. In addition, the equality ${\rm TC}(X)={\rm cat}(C_X)$ is proved in the (strict) metastable range $2p-1<q\leq3(p-1)$ under fairly mild conditions by making use of the Hopf invariant techniques recently developed by the authors in their study of the sectional category of arbitrar…

Inverting the Hopf map

We calculate the $η$-localization of the motivic stable homotopy ring over the complex numbers, confirming a conjecture of Guillou and Isaksen. Our approach is via the motivic Adams-Novikov spectral sequence. In fact, work of Hu, Kriz, and Ormsby implies that it suffices to compute the corresponding localization of the classical Adams-Novikov $E_2$-term, and this is what we do. Guillou and Isaksen also proposed a pattern of differentials in the localized motivic classical Adams spectral sequence, which we verify using a method first explored by Novikov.

The monochromatic Hopf invariant

In this paper we will compute the effect of the James-Hopf map after applying the Bousfield-Kuhn functor on Morava E-theory, and then compute the monochromatic Hopf invariant of the $β$ family using this cohomological information.