Exotic $$G_2$$-manifolds

Crowley, Diarmuid; Nordström, Johannes

doi:10.1007/s00208-020-02009-1

Cited by 5 publications

(8 citation statements)

References 32 publications

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“…By [, Theorem 1(ii)], see also Corollary , the underlying topological manifold admits a unique smooth structure. In , we find examples of manifolds with

G_{2}

holonomy where the smooth structure is not unique and calculating the generalised Eells–Kuiper invariant we find pairs of closed

G_{2}

‐manifolds that are homeomorphic but not diffeomorphic. For example

(M false(Z^{89}, 8 false), M false(Z^{89}, 8 false) ♯ {normalΣ}_{Mi})

is a pair of homeomorphic but not diffeomorphic smooth manifolds both of which admit metrics with

G_{2}

holonomy.…”

Section: Examplesmentioning

confidence: 94%

“…An important motivation for this paper is the study of Riemannian manifolds with holonomy the exceptional Lie group

G_{2}

: such manifolds always have

p_{M}

of infinite order (see Joyce [, Proposition 10.2.7]). In , we use the generalised Eells–Kuiper invariant to distinguish pairs of closed

G_{2}

‐manifolds which are homeomorphic but not diffeomorphic.…”

Section: Introductionmentioning

confidence: 99%

“…Proposition gives a similar way to compute the generalised Eells–Kuiper invariant via

{spin}^{c}

coboundaries. We use this method in to compute the generalised Eells–Kuiper invariants of certain closed 7‐manifolds with holonomy

G_{2}

that are homeomorphic but not diffeomorphic.…”

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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“…By [, Theorem 1(ii)], see also Corollary , the underlying topological manifold admits a unique smooth structure. In , we find examples of manifolds with

G_{2}

holonomy where the smooth structure is not unique and calculating the generalised Eells–Kuiper invariant we find pairs of closed

G_{2}

‐manifolds that are homeomorphic but not diffeomorphic. For example

(M false(Z^{89}, 8 false), M false(Z^{89}, 8 false) ♯ {normalΣ}_{Mi})

is a pair of homeomorphic but not diffeomorphic smooth manifolds both of which admit metrics with

G_{2}

holonomy.…”

Section: Examplesmentioning

confidence: 94%

“…An important motivation for this paper is the study of Riemannian manifolds with holonomy the exceptional Lie group

G_{2}

: such manifolds always have

p_{M}

of infinite order (see Joyce [, Proposition 10.2.7]). In , we use the generalised Eells–Kuiper invariant to distinguish pairs of closed

G_{2}

‐manifolds which are homeomorphic but not diffeomorphic.…”

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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“…Thus one obtains a configuration satisfying the hypotheses of Proposition 2.6. This is used in [3] and [4] to produce many examples of twisted connected sum G 2 -manifolds. Now consider the problem of finding matching bundles F ± → Z ± in order to construct G 2 -instantons by application of Theorem 1.2.…”

Section: ) and A Corresponds To An Ample Class On S For An Open Subcone Ampmentioning

confidence: 99%

“…To allow more possibilities, we want matchings r whose configuration N + , N − ⊂ L K3 has non-trivial intersection N 0 . Table 4 of [4] lists all 19 possible matchings of Fano 3-folds with Picard rank 2 such that N 0 is non-trivial. In this paper, the pair of building blocks we will consider comes from that table.…”

Section: ) and A Corresponds To An Ample Class On S For An Open Subcone Ampmentioning

confidence: 99%

Construction of $\mathrm{G}_2$-instantons via twisted connected sums

Menet

Nordström

Earp

2021

Mathematical Research Letters

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We propose a method to construct G 2 -instantons over a compact twisted connected sum G 2 -manifold, applying a gluing result of Sá Earp and Walpuski to instantons over a pair of 7-manifolds with a tubular end. In our example, the moduli spaces of the ingredient instantons are non-trivial, and their images in the moduli space over the asymptotic cross-section K3 surface intersect transversely. Such a pair of asymptotically stable holomorphic bundles is obtained using a twisted version of the Hartshorne-Serre construction, which can be adapted to produce other examples. Moreover, their deformation theory and asymptotic behaviour are explicitly understood, results which may be of independent interest.

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Extra-twisted connected sum $$G_2$$-manifolds

Nordström¹

2023

Ann Glob Anal Geom

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We present a construction of closed 7-manifolds of holonomy $$G_2$$ G 2 , which generalises Kovalev’s twisted connected sums by taking quotients of the pieces in the construction before gluing. This makes it possible to realise a wider range of topological types, and Crowley, Goette and the author use this to exhibit examples of closed 7-manifolds with disconnected moduli space of holonomy $$G_2$$ G 2 metrics.

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Exotic $$G_2$$-manifolds

Cited by 5 publications

References 32 publications

The classification of 2‐connected 7‐manifolds

The classification of 2‐connected 7‐manifolds

Construction of $\mathrm{G}_2$-instantons via twisted connected sums

Extra-twisted connected sum $$G_2$$-manifolds

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