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On $7$-manifolds with $b_{2}=2$: diffeomorphism classification and nonconnected moduli spaces of positive Ricci curvature metrics

Fupeng Xu

TL;DR

The paper classifies simply connected 7-manifolds with $b_{2}=2$ that arise as circle bundles over $N=(\mathbb{C}P^{1}\times\mathbb{C}P^{2})\#\mathbb{C}P^{3}$, using $s$-invariants and spin bordism to distinguish diffeomorphism types for the families $M_{m,n,l}$. It provides a coarse five-family classification, a detailed analysis for the intricate $\mathcal{M}_{1}$ family, and explicit $s$-invariant formulas that enable partial diffeomorphism classifications. The work computes $\Omega^{Spin}_{8}(K_{2})$ and $\Omega^{Spin}_{8}(K_{2};\mathrm{pr}_{1}^{*}\gamma^{1})$ and extends to $\mathcal{E}_{1}^{+}$-manifolds to define polarization-invariant data $s_{1},\varsigma,S_{10}$, essential for distinguishing diffeomorphism types in high dimensions. Finally, it applies these classifications to differential geometry, constructing metrics with positive Ricci curvature and proving the existence of manifolds whose space and moduli spaces of such metrics have infinitely many path components.

Abstract

We derive the $s$-invariants of certain simply connected $7$-manifolds whose second homology groups are isomorphic to $\mathbb{Z}^{2}$. We apply the $s$-invariants to give a partial classification of simply connected total spaces of circle bundles over $\left(\mathbb{C}P^{1}\times\mathbb{C}P^{2}\right)\#\mathbb{C}P^{3}$ up to diffeomorphism. As an application, we show that there is a simply connected $7$-manifold whose space and moduli space of positive Ricci curvature metrics both have infinitely many path components. We also determine bordism groups $Ω_{8}^{Spin}\left(K_{2}\right)$ and $Ω_{8}^{Spin}\left(K_{2};\mathrm{pr}_{1}^{*}γ^{1}\right)$ that are required in the deduction of $s$-invariants.

On $7$-manifolds with $b_{2}=2$: diffeomorphism classification and nonconnected moduli spaces of positive Ricci curvature metrics

TL;DR

The paper classifies simply connected 7-manifolds with that arise as circle bundles over , using -invariants and spin bordism to distinguish diffeomorphism types for the families . It provides a coarse five-family classification, a detailed analysis for the intricate family, and explicit -invariant formulas that enable partial diffeomorphism classifications. The work computes and and extends to -manifolds to define polarization-invariant data , essential for distinguishing diffeomorphism types in high dimensions. Finally, it applies these classifications to differential geometry, constructing metrics with positive Ricci curvature and proving the existence of manifolds whose space and moduli spaces of such metrics have infinitely many path components.

Abstract

We derive the -invariants of certain simply connected -manifolds whose second homology groups are isomorphic to . We apply the -invariants to give a partial classification of simply connected total spaces of circle bundles over up to diffeomorphism. As an application, we show that there is a simply connected -manifold whose space and moduli space of positive Ricci curvature metrics both have infinitely many path components. We also determine bordism groups and that are required in the deduction of -invariants.

Paper Structure

This paper contains 13 sections, 25 theorems, 206 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. A first classification of $M_{m,n,l}$
  3. Classification into five families $\mathcal{M}_{1}\sim\mathcal{M}_{5}$
  4. More information of manifolds in $\mathcal{M}_{1}$
  5. The $s$-invariant of spin manifolds in $\mathcal{M}_{1}$
  6. The bordism groups
  7. $s$-invariant for $\mathcal{E}_{1}^{+}$-manifolds
  8. Partial classification of manifolds in $\mathcal{M}_{1}$
  9. Computations of $s$-invariants for some spin manifolds in $\mathcal{M}_{1}$
  10. Partial classification of spin manifolds in $\mathcal{M}_{1}$
  11. Application to positive Ricci curvature problem
  12. Metrics $g_{m,n,l}$ and $h_{m,n,l}$ on $M_{m,n,l}$
  13. Infinitely many components of moduli space

Key Result

Theorem 1.1

Consider the following $5$ families of manifolds: If $M\in\mathcal{M}_{i}$, $\overline{M}\in\mathcal{M}_{j}$ and $1\leqslant i<j\leqslant5$, then $M$ and $\overline{M}$ are not homeomorphic.

Figures (3)

  • Figure 3.1: The $E^{2}$-page of Atiyah-Hirzebruch spectral sequence for $\Omega^{Spin}_{8}\left(K_{2}\right)$
  • Figure 3.2: $E_{\infty}$-page of Adams spectral sequence for $\Omega^{Spin}_{8}\left(K_{2}\right)$
  • Figure 3.3: Normal $2$-type and normal $2$-structure of $\mathcal{E}_{1}^{+}$-manifold $M$

Theorems & Definitions (56)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2: Complete classification of manifolds in $\mathcal{M}_{3}$
  • Theorem 1.3: Complete classification of manifolds in $\mathcal{M}_{2}$
  • Theorem 1.4: Partial classification of manifolds in $\mathcal{M}_{1}$
  • Theorem 1.5
  • Lemma 2.1
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.2
  • ...and 46 more