Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Coformality around fibrations and cofibrations

Ruizhi Huang

TL;DR

This work investigates how coformality behaves around fibrations and cofibrations in rational homotopy theory. It introduces the coformal limit $X'$ to capture purely quadratic models and proves a dual to Lupton's formality results: under TNHZ, coformal fiber and base, and $cat_0(E')\le 2$, the total space is coformal; it also establishes a cofibration analogue for inert attaching maps. The results extend to Koszul properties of total spaces in certain spherical fibrations, linking coformality with Koszul geometry. The paper further analyzes the necessity and sharpness of the hypotheses, providing counterexamples that delineate the boundaries of these phenomena.

Abstract

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}.

Coformality around fibrations and cofibrations

TL;DR

This work investigates how coformality behaves around fibrations and cofibrations in rational homotopy theory. It introduces the coformal limit to capture purely quadratic models and proves a dual to Lupton's formality results: under TNHZ, coformal fiber and base, and , the total space is coformal; it also establishes a cofibration analogue for inert attaching maps. The results extend to Koszul properties of total spaces in certain spherical fibrations, linking coformality with Koszul geometry. The paper further analyzes the necessity and sharpness of the hypotheses, providing counterexamples that delineate the boundaries of these phenomena.

Abstract

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}.

Paper Structure

This paper contains 13 sections, 12 theorems, 42 equations.

Table of Contents

  1. Background
  2. Coformality
  3. Koszul property
  4. Organization of the paper
  5. Preliminary results
  6. Two preliminary lemmas
  7. A simple case
  8. Coformal limit
  9. Proof of of main theorems
  10. Proof of Theorem \ref{['coformalmainthm']}
  11. Proof of Theorem \ref{['coformalmainthm2']}
  12. Proof of Theorem \ref{['fibkoszulthm2']}
  13. Discussions on the conditions of Theorem \ref{['coformalmainthm']}

Key Result

Theorem 1

Let be a fibration of simply connected topological spaces of finite type. Suppose the fibration is TNHZ, and $F$ is coformal. If $B$ is coformal and $cat_0(E^\prime)\leq 2$, then $E$ is coformal.

Theorems & Definitions (30)

  • Theorem 1
  • Example 2
  • Theorem 3
  • Example 4
  • Example 5
  • Proposition 6
  • Theorem 7
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • ...and 20 more