A note on the Petersen-Wilhelm conjecture

González-Álvaro, David; Radeschi, Marco

doi:10.1090/proc/14070

Cited by 5 publications

(10 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The conjecture was confirmed for the known simply-connected even-dimensional examples of positive curvature in [4] and for the known simply-connected odddimensional examples in [31]. It seems that this conjecture is very geometrical in nature lacking the connection to topology on which we focus in this survey.…”

Section: Conjecture 218 (Dessai) Let (M G) Be a Spin Manifold Admitmentioning

confidence: 53%

Bounds on Sectional Curvature and Interactions with Topology

Amann

2020

Jahresber. Dtsch. Math. Ver.

View full text Add to dashboard Cite

In this survey article we exemplarily illustrate implications of curvature assumptions on the topology of the underlying manifold. We shall mainly focus on sectional curvature and three different kinds of restrictions, namely on non-negative respectively on positive sectional curvature, as well as on two-sided curvature bounds.We shall see that there are various implications on the side of topology, namely, for example, geometry having an impact on elementary invariants like the Euler characteristic or Betti numbers as well as on concepts from rational homotopy theory or index theory, and that there are connections to K-theory.On our way of making these connections we shall draw on certain simplifications and tools like group actions or metrics with additional properties like geometric formality.

show abstract

Section: Conjecture 218 (Dessai) Let (M G) Be a Spin Manifold Admitmentioning

confidence: 53%

Bounds on Sectional Curvature and Interactions with Topology

Amann

2020

Jahresber. Dtsch. Math. Ver.

View full text Add to dashboard Cite

show abstract

“…The known examples of simply-connected positively curved closed manifolds are the following (cf. [7]):…”

Section: These Classes Of Examples Enrich Theorem Dmentioning

confidence: 99%

“…Without going into details, collecting the information for example from [23], [1], [7] we derive that all these spaces are formal and elliptic, and, in any case, the following holds:…”

Section: These Classes Of Examples Enrich Theorem Dmentioning

confidence: 99%

“…We derive that h(X) < h(F ) + dim π * (B) ⊗ Q 2 (dim π * (B)⊗Q)/2+1 (11) and we need to discuss the inequality n 2 n/2+1 ≤ 1 4 for n ∈ N (playing the role of dim π * (B) ⊗ Q). This holds true unless n ∈ [1,7]. So, in order to finish the proof, we need to differ and discuss the following particular cases:…”

Section: Proofs Of Theorems D and Ementioning

confidence: 99%

“…In [1] we proved the Petersen-Wilhelm conjecture in a much more general context for the known examples of positive curvature of even dimensions only using their rational structure. (Since several odd-dimensional examples rationally split as a product, rational tools are not enough in odd dimensions; in [7] the odd-dimensional examples are verified using finite coefficients. )…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Homology versus homotopy in fibrations and in limits

Amann

2020

Preprint

View full text Add to dashboard Cite

Motivated by prominent problems like the Hilali conjecture Yamaguchi-Yokura recently proposed certain estimates on the relations of the dimensions of rational homotopy and rational cohomology groups of fibre, base and total spaces in a fibration of rationally elliptic spaces.In this article we prove these estimates in the category of formal elliptic spaces and, in general, whenever the total space in addition has positive Euler characteristic or has the rational homotopy type of a homogeneous manifold (respectively of a known example) of positive sectional curvature. Additionally, we provide general estimates approximating the conjectured ones.Moreover, we suggest to study families of rationally elliptic spaces under certain asymptotics, and we discuss the conjectured estimates from this perspective for two-stage spaces.

show abstract

Infinite families of manifolds of positive $$k\mathrm{th}$$-intermediate Ricci curvature with k small

2022

Self Cite

View full text Add to dashboard Cite

Positive $$k\mathrm{th}$$ k th -intermediate Ricci curvature on a Riemannian n-manifold, to be denoted by $${{\,\mathrm{Ric}\,}}_k>0$$ Ric k > 0 , is a condition that interpolates between positive sectional and positive Ricci curvature (when $$k =1$$ k = 1 and $$k=n-1$$ k = n - 1 respectively). In this work, we produce many examples of manifolds of $${{\,\mathrm{Ric}\,}}_k>0$$ Ric k > 0 with k small by examining symmetric and normal homogeneous spaces, along with certain metric deformations of fat homogeneous bundles. As a consequence, we show that every dimension $$n\ge 7$$ n ≥ 7 congruent to $$3\,{{\,\mathrm{mod}\,}}4$$ 3 mod 4 supports infinitely many closed simply connected manifolds of pairwise distinct homotopy type, all of which admit homogeneous metrics of $${{\,\mathrm{Ric}\,}}_k>0$$ Ric k > 0 for some $$k<n/2$$ k < n / 2 . We also prove that each dimension $$n\ge 4$$ n ≥ 4 congruent to 0 or $$1\,{{\,\mathrm{mod}\,}}4$$ 1 mod 4 supports closed manifolds which carry metrics of $${{\,\mathrm{Ric}\,}}_k>0$$ Ric k > 0 with $$k\le n/2$$ k ≤ n / 2 , but do not admit metrics of positive sectional curvature.

show abstract

A note on the Petersen-Wilhelm conjecture

Cited by 5 publications

References 23 publications

Bounds on Sectional Curvature and Interactions with Topology

Bounds on Sectional Curvature and Interactions with Topology

Homology versus homotopy in fibrations and in limits

Infinite families of manifolds of positive $$k\mathrm{th}$$-intermediate Ricci curvature with k small

Contact Info

Product

Resources

About