New invariants of<i>G</i><sub>2</sub>–structures

Crowley, Diarmuid; Nordström, Johannes

doi:10.2140/gt.2015.19.2949

Cited by 53 publications

(49 citation statements)

References 32 publications

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“…2 t and q h , which have equal inhomogeneity β h = p − 2h by definition, also have equal Arf invariant by (11). Therefore, by Theorem 2.16 there is an automorphism…”

Section: The Action Of Aut Q • On Gauss Refinementsmentioning

confidence: 84%

“…By the Chinese remainder theorem, these two constraints completely characterise Δ(k, t) as an element of Q/2d π Z. Now observe that A(q eπ(k+t) ) = A(q eπk −eπt ) = A(q eπk ) − q eπk (−e π t) ∈ Q/Z by 2.18(ii) and (11). Thus (17b) is equivalent to requiring that g(k) − A(q eπk ) mod Z is constant and that (20) holds.…”

Section: Gauss Refinementsmentioning

confidence: 90%

“…For other problems, for example counting the number of deformation equivalence classes of

G_{2}

‐structures on

M

as in , it is important to know the value of

p^{2}

for diffeomorphisms. Hence, we defined the smooth reactivity of

M

R^{Diff} false(M false)

, and the smooth ( co ) homologically fixed reactivity of

M

R_{H}^{Diff} false(M false)

by the equations

p^{2} false({\tilde{π}}_{0} Diff (M) false) = R^{Diff} false(M false) Z and p^{2} false({\tilde{π}}_{0} {Diff}_{H} (M) false) = R_{H}^{Diff} false(M false) Z,

where

{\overset{π}{true}}_{0} {Diff}_{H} false(M false) \subseteq {\overset{π}{true}}_{0} Diff false(M false)

is the subgroup of pseudo‐isotopy classes acting trivially on

H^{*} false(M false)

.…”

Section: The Classification Of 2‐connected 7‐manifoldsmentioning

confidence: 99%

“…Since the image of p 2 plays a key role, we define nonnegative integers called the reactivity of M , R(M ), and the (co)homologically fixed reactivity of M , R H (M ), by the following equations For other problems, for example counting the number of deformation equivalence classes of G 2 -structures on M as in [11], it is important to know the value of p 2 for diffeomorphisms. Hence, we defined the smooth reactivity of M , R Diff (M ), and the smooth (co)homologically fixed reactivity of M , R Diff H (M ) by the equations…”

Section: Inertia and Reactivity In More Detailmentioning

confidence: 99%

“…The computation of the reactivity of

M

also has applications in

G_{2}

‐topology. We define the smooth reactivity of

M

R^{Diff} false(M false)

, using the equation

p^{2} false(Diff (M) false) = R^{Diff} false(M false) Z

, and in [, Section 6] we show that

R^{Diff} false(M false)

determines the number of

G_{2}

‐structures on

M

modulo homotopies and diffeomorphisms. By Corollary ,

R^{Diff} false(M false) = lcm false(2^{r} d_{o}, 224 false)

for 2‐connected

M

, and this allows us to generalise Theorem to give a classification of 2‐connected 7‐manifolds equipped with a

G_{2}

‐structure, up to diffeomorphisms and homotopies of

G_{2}

‐structures [, Theorem 6.9].…”

Section: Introductionmentioning

confidence: 99%

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The classification of 2‐connected 7‐manifolds

Crowley

Nordström

2018

Proc. London Math. Soc.

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We present a comprehensive classification of closed smooth 2‐connected manifolds of dimension 7. This builds on the almost‐smooth classification from the first author's thesis. The main new ingredient is a generalisation of the Eells–Kuiper invariant that is defined for any closed spin 7‐manifold M, regardless of whether the spin characteristic class pM∈H4false(Mfalse) is torsion. We also determine the inertia group of 2‐connected M — equivalently the number of oriented smooth structures on the underlying topological manifold — in terms of pM and the torsion linking form.

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“…2 t and q h , which have equal inhomogeneity β h = p − 2h by definition, also have equal Arf invariant by (11). Therefore, by Theorem 2.16 there is an automorphism…”

Section: The Action Of Aut Q • On Gauss Refinementsmentioning

confidence: 84%

Section: Gauss Refinementsmentioning

confidence: 90%

“…For other problems, for example counting the number of deformation equivalence classes of

G_{2}

‐structures on

M

as in , it is important to know the value of

p^{2}

for diffeomorphisms. Hence, we defined the smooth reactivity of

M

R^{Diff} false(M false)

, and the smooth ( co ) homologically fixed reactivity of

M

R_{H}^{Diff} false(M false)

by the equations

p^{2} false({\tilde{π}}_{0} Diff (M) false) = R^{Diff} false(M false) Z and p^{2} false({\tilde{π}}_{0} {Diff}_{H} (M) false) = R_{H}^{Diff} false(M false) Z,

where

{\overset{π}{true}}_{0} {Diff}_{H} false(M false) \subseteq {\overset{π}{true}}_{0} Diff false(M false)

is the subgroup of pseudo‐isotopy classes acting trivially on

H^{*} false(M false)

.…”

Section: The Classification Of 2‐connected 7‐manifoldsmentioning

confidence: 99%

Section: Inertia and Reactivity In More Detailmentioning

confidence: 99%

“…The computation of the reactivity of

M

also has applications in

G_{2}

‐topology. We define the smooth reactivity of

M

R^{Diff} false(M false)

, using the equation

p^{2} false(Diff (M) false) = R^{Diff} false(M false) Z

, and in [, Section 6] we show that

R^{Diff} false(M false)

determines the number of

G_{2}

‐structures on

M

modulo homotopies and diffeomorphisms. By Corollary ,

R^{Diff} false(M false) = lcm false(2^{r} d_{o}, 224 false)

for 2‐connected

M

, and this allows us to generalise Theorem to give a classification of 2‐connected 7‐manifolds equipped with a

G_{2}

‐structure, up to diffeomorphisms and homotopies of

G_{2}

‐structures [, Theorem 6.9].…”

Section: Introductionmentioning

confidence: 99%

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The classification of 2‐connected 7‐manifolds

Crowley

Nordström

2018

Proc. London Math. Soc.

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Purely coclosed G₂‐structures on nilmanifolds

Bazzoni

Garvı́n

Muñoz

2023

Mathematische Nachrichten

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We classify seven‐dimensional nilpotent Lie groups, decomposable or of nilpotency step at most 4, endowed with left‐invariant purely coclosed G2‐structures. This is done by going through the list of all seven‐dimensional nilpotent Lie algebras given by Gong, providing an example of a left‐invariant 3‐form φ which is a pure coclosed G2‐structure (i.e., it satisfies d∗φ=0$d*\varphi =0$, φ∧dφ=0$\varphi \wedge d\varphi =0$) for those nilpotent Lie algebras that admit them; and by showing the impossibility of having a purely coclosed G2‐structure for the rest of them.

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Supersymmeric Gauge Theory on Curved 7‐Branes

Prins

2019

Fortschritte der Physik

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We construct novel 7d supersymmetric gauge theories which include a Chern‐Simons‐like term on curved spaces. In order to so, we examine the supersymmetry constraints for E7‐branes in type IIA* theory, rather than making use of an off‐shell supergravity. We find two classes of solutions to the constraints, expressed in terms of a G2‐structure with non‐vanishing intrinsic torsion. The supersymmetric gauge theories are then obtained by coupling flat gauge theory to such supersymmetric backgrounds. Various examples are given, including round and squashed S7 as well as S3×M4 with M4 hyperkähler.

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New invariants ofG₂–structures

Cited by 53 publications

References 32 publications

The classification of 2‐connected 7‐manifolds

The classification of 2‐connected 7‐manifolds

Purely coclosed G₂‐structures on nilmanifolds

Supersymmeric Gauge Theory on Curved 7‐Branes

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New invariants ofG2–structures

Cited by 53 publications

References 32 publications

The classification of 2‐connected 7‐manifolds

The classification of 2‐connected 7‐manifolds

Purely coclosed G2‐structures on nilmanifolds

Supersymmeric Gauge Theory on Curved 7‐Branes

Contact Info

Product

Resources

About

New invariants ofG₂–structures

Purely coclosed G₂‐structures on nilmanifolds