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Moduli Spaces of Positive Curvature Metrics in Dimension Four and Beyond

Thorsten Hertl

TL;DR

The work investigates higher-homotopy structure in observer moduli spaces of curvature-constrained metrics on complex projective spaces and related four-manifolds. It develops foundational models for moduli and observer moduli spaces, introduces a nontrivial fibre bundle DO(1) built from a twisted Hopf fibration, and constructs fibrewise metrics with positive scalar or sectional curvature to detect nontrivial elements. The main contributions are: (i) explicit nontrivial elements in $\pi_2(\ObsModuli[\mathbb{C}P^n][\mathrm{sec}>0],[g_{FS}])$ with orders depending on parity of $n$ and nontrivial higher rational homotopy groups, (ii) nontrivial PSC-observer classes arising from gluing twisted disc bundles to four-manifold copies and using $p_1$-intersections to obstruct trivializations, and (iii) a parallel construction for positive sectional curvature on twisted projective bundles $\mathbb{C}P^n_{\Phi}$ giving finite-order elements in $\pi_2$ and nontrivial rational higher homotopy. The results extend the known nontriviality of observer moduli spaces beyond small-dimensional or spin-dependent cases, highlighting new geometric and topological features of positively curved metrics.

Abstract

We construct non-trivial elements in the homotopy groups of the observer moduli space of positive sectional curvature metrics on $\mathbb{C}P^n$ and non-trivial elements in the homotopy groups of the observer moduli space of positive scalar curvature metrics on $\mathbb{C}P^2 \sharp M^4$.

Moduli Spaces of Positive Curvature Metrics in Dimension Four and Beyond

TL;DR

The work investigates higher-homotopy structure in observer moduli spaces of curvature-constrained metrics on complex projective spaces and related four-manifolds. It develops foundational models for moduli and observer moduli spaces, introduces a nontrivial fibre bundle DO(1) built from a twisted Hopf fibration, and constructs fibrewise metrics with positive scalar or sectional curvature to detect nontrivial elements. The main contributions are: (i) explicit nontrivial elements in with orders depending on parity of and nontrivial higher rational homotopy groups, (ii) nontrivial PSC-observer classes arising from gluing twisted disc bundles to four-manifold copies and using -intersections to obstruct trivializations, and (iii) a parallel construction for positive sectional curvature on twisted projective bundles giving finite-order elements in and nontrivial rational higher homotopy. The results extend the known nontriviality of observer moduli spaces beyond small-dimensional or spin-dependent cases, highlighting new geometric and topological features of positively curved metrics.

Abstract

We construct non-trivial elements in the homotopy groups of the observer moduli space of positive sectional curvature metrics on and non-trivial elements in the homotopy groups of the observer moduli space of positive scalar curvature metrics on .
Paper Structure (5 sections, 20 theorems, 78 equations)

This paper contains 5 sections, 20 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Foundations on Moduli Spaces
  3. The Construction of the bundle $\mathcal{DO}(1)$
  4. Proof of Theorem \ref{['Main Thm - PSC']} and \ref{['Main Thm - PSC with order']}
  5. Proof of Theorem \ref{['Main Thm - PSec']}

Key Result

Theorem 1

Let $n \geq 2$ and let $3 \leq j \leq n-1$ be odd. Let further $g_{FS}$ be the Fubini-Study metric on $\mathord{\mathbb{C}P}^n$. Then we have the following results: Moreover, if $C$ is an open, diffeomorphism invariant curvature condition that is implied by positive sectional curvature, then the same results hold for $\mathrm{Riem}^{C}(\mathord{\mathbb{C}P}^n)/\mathrm{Diff}_{x_0}(\mathord{\mathbb

Theorems & Definitions (49)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 39 more