Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe Rational Sectional Category of Certain Universal Fibrations
64
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We prove that the sectional category of the universal fibration with fibre X, for X any space that satisfies a well-known conjecture of Halperin, equals one after rationalization.
rate research
Read More
Let $X$ be a simply connected space with finite-dimensional rational homotopy groups. Let $p_infty colon UE to mathrm{Baut}_1(X)$ be the universal fibration of simply connected spaces with fibre $X$. We give a DG Lie model for the evaluation map $ omega colon mathrm{aut}_1(mathrm{Baut}_1(X_{mathbb Q})) to mathrm{Baut}_1(X_{mathbb Q})$ expressed in terms of derivations of the relative Sullivan model of $p_infty$. We deduce formulas for the rational Gottlieb group and for the evaluation subgroups of the classifying space $mathrm{Baut}_1(X_{mathbb Q})$ as a consequence. We also prove that ${mathbb C} P^n_{mathbb Q}$ cannot be realized as $mathrm{Baut}_1(X_{mathbb Q})$ for $n leq 4$ and $X$ with finite-dimensional rational homotopy groups.
A fibration of ${mathbb R}^n$ by oriented copies of ${mathbb R}^p$ is called skew if no two fibers intersect nor contain parallel directions. Conditions on $p$ and $n$ for the existence of such a fibration were given by Ovsienko and Tabachnikov. A classification of smooth fibrations of ${mathbb R}^3$ by skew oriented lines was given by Salvai, in analogue with the classification of oriented great circle fibrations of $S^3$ by Gluck and Warner. We show that Salvais classification has a topological variation which generalizes to characterize all continuous fibrations of ${mathbb R}^n$ by skew oriented copies of ${mathbb R}^p$. We show that the space of fibrations of ${mathbb R}^3$ by skew oriented lines deformation retracts to the subspace of Hopf fibrations, and therefore has the homotopy type of a pair of disjoint copies of $S^2$. We discuss skew fibrations in the complex and quaternionic setting and give a necessary condition for the existence of a fibration of ${mathbb C}^n$ (${mathbb H}^n$) by skew oriented copies of ${mathbb C}^p$ (${mathbb H}^p$).
We study the structure of the rational motivic stable homotopy category over general base schemes. Our first class of results concerns the six operations: we prove absolute purity, stability of constructible objects, and Grothendieck-Verdier duality for SH_Q. Next, we prove that SH_Q is canonically SL-oriented; we compare SH_Q with the category of rational Milnor-Witt motives; and we relate the rational bivariant A^1-theory to Chow-Witt groups. These results are derived from analogous statements for the minus part of SH[1/2].
We show that in a fibration the coformality of the base space implies the coformality of the total space under reasonable conditions, and these conditions can not be weakened. The result is partially dual to the classical work of Lupton cite{Lup} on the formality within a fibration. Our result has two applications. First, we show that for certain cofibrations, the coformality of the cofiber implies the coformality of the base. Secondly, we show that the total spaces of certain spherical fibrations are Koszul in the sense of Berglund cite{Ber}.
We consider the topological category of $h$-cobordisms between manifolds with boundary and compare its homotopy type with the standard $h$-cobordism space of a compact smooth manifold.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
