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Topological properties of closed $\widetilde{\mathrm{G}}_2$, $\mathrm{SL}(3;\mathbb{C})$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms on manifolds

Laurence H. Mayther

TL;DR

This work develops a framework for the topological study of closed $\sigma_0$-forms for four stable forms on oriented manifolds using algebro-topological methods and $h$-principles. It proves that an oriented 7-manifold admits a closed $\widetilde{\mathrm{G}}_2$-structure iff it is spin, and for spin manifolds every degree-$3$ and degree-$4$ cohomology class can be represented by closed $\widetilde{\mathrm{G}}_2$ 3-forms and 4-forms; it introduces a generalized $\widetilde{\mathrm{G}}_2$-cobordism extending Donaldson's notion to $\widetilde{\mathrm{G}}_2$, $\mathrm{SL}(3;\mathbb{C})$, and $\mathrm{SL}(3;\mathbb{R})^2$ forms, showing homotopic $\mathrm{SL}(3;\mathbb{C})$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms in a fixed cohomology class are cobordant. The paper then provides a complete homotopy classification of closed $\mathrm{SL}(3;\mathbb{C})$ forms on 6-manifolds and a lower bound on the number of homotopy classes for closed $\mathrm{SL}(3;\mathbb{R})^2$ forms, together with extendibility criteria linking Euler class and $w_2$-data. Finally, extendibility questions for boundary values are explored, tying the topology of underlying manifolds to the existence and boundary behavior of these special forms.

Abstract

This paper uses algebro-topological techniques such as characteristic classes and obstruction theory, together with the $h$-principles for $\widetilde{\mathrm{G}}_2$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms recently established by the author and the $h$-principle for $\mathrm{SL}(3;\mathbb{C})$ forms established by Donaldson, to prove results on the topological properties of closed $\widetilde{\mathrm{G}}_2$, $\mathrm{SL}(3;\mathbb{C})$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms on oriented 6- and 7-manifolds. Specifically, a criterion for an arbitrary oriented 7-manifold to admit a closed (resp. coclosed) $\widetilde{\mathrm{G}}_2$-structure is obtained, proving a conjecture of Lê; a generalisation of Donaldson's '$\mathrm{G}_2$-cobordisms' to $\widetilde{\mathrm{G}}_2$, $\mathrm{SL}(3;\mathbb{C})$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms is introduced, with homotopic $\mathrm{SL}(3;\mathbb{C})$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms in a given cohomology class shown to be $\widetilde{\mathrm{G}}_2$-cobordant, a result which currently has no analogue in the $\mathrm{G}_2$ case; and a complete classification of closed $\mathrm{SL}(3;\mathbb{C})$ forms up to homotopy is provided. Additionally, a lower bound on the number of homotopy classes of closed $\mathrm{SL}(3;\mathbb{R})^2$ forms on a given manifold is obtained, and the question of which closed $\mathrm{SL}(3;\mathbb{C})$ or $\mathrm{SL}(3;\mathbb{R})^2$ forms arise as the boundary values of closed $\widetilde{\mathrm{G}}_2$-structures on oriented 7-manifolds is investigated.

Topological properties of closed $\widetilde{\mathrm{G}}_2$, $\mathrm{SL}(3;\mathbb{C})$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms on manifolds

TL;DR

This work develops a framework for the topological study of closed -forms for four stable forms on oriented manifolds using algebro-topological methods and -principles. It proves that an oriented 7-manifold admits a closed -structure iff it is spin, and for spin manifolds every degree- and degree- cohomology class can be represented by closed 3-forms and 4-forms; it introduces a generalized -cobordism extending Donaldson's notion to , , and forms, showing homotopic and forms in a fixed cohomology class are cobordant. The paper then provides a complete homotopy classification of closed forms on 6-manifolds and a lower bound on the number of homotopy classes for closed forms, together with extendibility criteria linking Euler class and -data. Finally, extendibility questions for boundary values are explored, tying the topology of underlying manifolds to the existence and boundary behavior of these special forms.

Abstract

This paper uses algebro-topological techniques such as characteristic classes and obstruction theory, together with the -principles for and forms recently established by the author and the -principle for forms established by Donaldson, to prove results on the topological properties of closed , and forms on oriented 6- and 7-manifolds. Specifically, a criterion for an arbitrary oriented 7-manifold to admit a closed (resp. coclosed) -structure is obtained, proving a conjecture of Lê; a generalisation of Donaldson's '-cobordisms' to , and forms is introduced, with homotopic and forms in a given cohomology class shown to be -cobordant, a result which currently has no analogue in the case; and a complete classification of closed forms up to homotopy is provided. Additionally, a lower bound on the number of homotopy classes of closed forms on a given manifold is obtained, and the question of which closed or forms arise as the boundary values of closed -structures on oriented 7-manifolds is investigated.
Paper Structure (3 sections, 7 theorems, 9 equations)

This paper contains 3 sections, 7 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Elementary properties of $\hbox{\boldmath $\widetilde{\mathrm{G}}_2$}$, $\hbox{\boldmath $\operatorname{SL}(3;\mathbb{C})$}$ and $\hbox{\boldmath $\operatorname{SL}(3;\mathbb{R})^2$}$ forms

Key Result

Theorem 1.3

Let $\mathrm{M}$ be an oriented 7-manifold (not necessarily closed and possibly with boundary). Then, $\mathrm{M}$ admits $\widetilde{\mathrm{G}}_2$-structures if and only if it is spin.

Theorems & Definitions (7)

  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9