Formality of $k$-connected spaces in $4k+3$ and $4k+4$ dimensions

Cavalcanti, Gil R.

doi:10.1017/s0305004106009340

Cited by 7 publications

(10 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For the special case of twisted connected sums, one may ask whether formality can be deduced from the way they are built from pairs of Kähler manifolds. Cavalcanti [13] shows that any simply-connected 7-manifold M with b 2 (M ) ≤ 1 is formal (so we did not need to consider Massey products when π 2 (M ) is cyclic), and that b 2 (M ) ≤ 2 suffices for formality if M is a G 2 -manifold. Formality of G 2 -manifolds is also studied by Verbitsky [71].…”

Section: Review and Outlookmentioning

confidence: 99%

G 2 -manifolds and associative submanifolds via semi-Fano 3 -folds

Corti¹,

Haskins²,

Nordström³

et al. 2015

Duke Math. J.

131

318

View full text Add to dashboard Cite

Abstract. We provide a significant extension of the twisted connected sum construction of G2-manifolds, ie Riemannian 7-manifolds with holonomy group G2, first developed by Kovalev; along the way we address some foundational questions at the heart of the twisted connected sum construction. Our extension allows us to prove many new results about compact G2-manifolds and leads to new perspectives for future research in the area. Some of the main contributions of the paper are:(i) We correct, clarify and extend several aspects of the K3 "matching problem" that occurs as a key step in the twisted connected sum construction. (ii) We show that the large class of asymptotically cylindrical Calabi-Yau 3-folds built from semi-Fano 3-folds (a subclass of weak Fano 3-folds) can be used as components in the twisted connected sum construction. (iii) We construct many new topological types of compact G2-manifolds by applying the twisted connected sum to asymptotically Calabi-Yau 3-folds of semi-Fano type studied in [18]. (iv) We obtain much more precise topological information about twisted connected sum G2-manifolds; one application is the determination for the first time of the diffeomorphism type of many compact G2-manifolds. (v) We describe "geometric transitions" between G2-metrics on different 7-manifolds mimicking "flopping" behaviour among semi-Fano 3-folds and "conifold transitions" between Fano and semi-Fano 3-folds. (vi) We construct many G2-manifolds that contain rigid compact associative 3-folds. (vii) We prove that many smooth 2-connected 7-manifolds can be realised as twisted connected sums in numerous ways; by varying the building blocks matched we can vary the number of rigid associative 3-folds constructed therein. The latter result leads to speculation that the moduli space of G2-metrics on a given 7-manifold may consist of many different connected components, and opens up many further questions for future study. For instance, the higher-dimensional gauge theory invariants proposed by Donaldson may provide ways to detect G2-metrics on a given 7-manifold that are not deformation equivalent.

show abstract

Section: Review and Outlookmentioning

confidence: 99%

G 2 -manifolds and associative submanifolds via semi-Fano 3 -folds

Corti¹,

Haskins²,

Nordström³

et al. 2015

Duke Math. J.

131

318

View full text Add to dashboard Cite

show abstract

“…Cavalcanti [, Theorem 4] showed that if

M

is a closed

(n - 1)

‐connected

(4 n - 1)

‐manifold and there is an element

φ \in H^{2 n - 1} false(M false)

such that

H^{n} false(M false) \to H^{3 n - 1} false(M false)

x \mapsto φ x

is an isomorphism (‘

M

has a hard Lefschetz property') then

M

is formal if

b_{n} false(M false) ⩽ 2

and its Massey products vanish uniformly if

b_{n} false(M false) ⩽ 3

. As an illustration of our results we can make the following improvement.…”

Section: Introductionmentioning

confidence: 99%

The rational homotopy type of (n−1)‐connected manifolds of dimension up to 5n−3

Crowley

Nordström

2020

Journal of Topology

View full text Add to dashboard Cite

We define the Bianchi-Massey tensor of a topological space X to be a linear map B → H * (X), where B is a subquotient of H * (X) ⊗4 determined by the algebra H * (X). We then prove that if M is a closed (n−1)-connected manifold of dimension at most 5n−3 (and n ≥ 2) then its rational homotopy type is determined by its cohomology algebra and Bianchi-Massey tensor, and that M is formal if and only if the Bianchi-Massey tensor vanishes.We use the Bianchi-Massey tensor to show that there are many (n−1)-connected (4n−1)manifolds that are not formal but have no non-zero Massey products, and to present a classification of simply-connected 7-manifolds up to finite ambiguity. 1 arXiv:1505.04184v2 [math.AT]

show abstract

“…It would be interesting to construct A ∞ -minimal models for other types of dgas or A ∞ -algebras following this way. For example, we may be able to extend Cavalcanti's result that a compact orientable k-connected manifold of dimension 4k + 3 or 4k + 4 with b k+1 = 1 is formal [3].…”

Section: Two Conjecturesmentioning

confidence: 80%

$A_\infty$-Minimal Model on Differential Graded Algebras

Zhou¹

2019

Preprint

View full text Add to dashboard Cite

For a formal differential graded algebra, if extended by an odd degree element, we prove that the extended algebra has an A ∞ -minimal model with only m 2 and m 3 non-trivial. As an application, the A ∞ -algebras constructed by Tsai, Tseng and Yau on formal symplectic manifolds satisfy this property. Separately, we expand the result of Miller and Crowley-Nordström for k-connected manifold. In particular, we prove that if the dimension of the manifold n ≤ (l +1)k +2, then its de Rham complex has an A ∞ -minimal model with m p = 0 for all p ≥ l.

show abstract

Formality of $k$-connected spaces in $4k+3$ and $4k+4$ dimensions

Cited by 7 publications

References 20 publications

G 2 -manifolds and associative submanifolds via semi-Fano 3 -folds

G 2 -manifolds and associative submanifolds via semi-Fano 3 -folds

The rational homotopy type of (n−1)‐connected manifolds of dimension up to 5n−3

$A_\infty$-Minimal Model on Differential Graded Algebras

Contact Info

Product

Resources

About