The cup-length of the oriented Grassmannians vs a new bound for zero-cobordant manifolds

Korbaš, Július

doi:10.36045/bbms/1267798499

Cited by 19 publications

(45 citation statements)

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“…Corollary 3.8) we show that the same upper bound will also work for the whole interval [2 t−1 + 2 t−1 3 , 2 t − 2]. We also provide a lower bound for cup( G n, 3 ). These bounds are consistent with Fukaya's conjecture.…”

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confidence: 53%

“…We have that π * (w 1 ) = 0, w 2 := π * (w 2 ) = 0, w 3 := π * (w 3 ) = 0. 3 ) consists of all the homogeneous polynomials of degree j built out of w 2 and w 3 . As n ≥ 6, j = 2 then j + 1 is always less than n − 2.…”

Section: Cohomology Rings Of Relevant Spaces 21 Preliminariesmentioning

confidence: 99%

“…Let w 2 := π * (w 2 ), w 3 := π * (w 3 ) denote the 2nd and 3rd Stiefel-Whitney classes of γ n, 3 , which is the pullback of γ n,3 via the covering map π : G n,3 → G n, 3 . We shall call this bundle the oriented tautological bundle.…”

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confidence: 99%

“…Recall that Z 2 -cup-length of a path-connected space X, denoted by cup(X), is defined as the maximum r such that there exist classes x 1 , x 2 , · · · , x r ∈ H * (X; Z 2 ) with non-trivial cup product, i.e., x 1 ∪ x 2 ∪ · · · ∪ x r = 0. Fukaya in [2, Conjecture 1.2] conjectured about the Z 2 -cup-length of oriented Grassmann manifolds G n, 3 . In [2,4,6,12] the authors have proved the conjecture for n = 2 t − 1, 2 t , 2 t + 1, 2 t + 2, 2 t + 2 t−1 + 1, 2 t + 2 t−1 + 2 (t ≥ 3).…”

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confidence: 99%

“…Let t ≥ 4 and n ∈ [2 t−1 + 2 t− 1 3 , 2 t − 2]. Then the Z 2 -cup-length cup( G n,3 ) is bounded above by 3(n−3) In another direction we have studied characteristic rank of G n,k (k ≥ 5) and upper characteristic rank of G n, 3 . If X is a connected finite CW complex and ξ is a real (finite rank) vector bundle over X, recall from [10] that the characteristic rank of ξ over X, denoted by charrank X (ξ), is by definition the largest integer k ≤ dim(X) such that every cohomology class x ∈ H j (X; Z 2 ), 0 ≤ j ≤ k, is a polynomial in the Stiefel-Whitney classes w i (ξ).…”

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confidence: 99%

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On the cohomology ring and upper characteristic rank of Grassmannian of oriented 3-planes

Basu

Chakraborty

2019

J. Homotopy Relat. Struct.

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In this paper we study the mod 2 cohomology ring of the Grasmannian Gn,3 of oriented 3-planes in R n . We determine the degrees of the indecomposable elements in the cohomology ring. We also obtain an almost complete description of the cohomology ring. This partial description allows us to provide lower and upper bounds on the cup length of Gn,3. As another application, we show that the upper characteristic rank of Gn,3 equals the characteristic rank of γn,3, the oriented tautological bundle over Gn,3.

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confidence: 53%

Section: Cohomology Rings Of Relevant Spaces 21 Preliminariesmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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On the cohomology ring and upper characteristic rank of Grassmannian of oriented 3-planes

Basu

Chakraborty

2019

J. Homotopy Relat. Struct.

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Characteristic rank of vector bundles over Stiefel manifolds

2012

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The characteristic rank of a vector bundle $\xi$ over a finite connected $CW$-complex $X$ is by definition the largest integer $k$, $0\leq k\leq \mathrm{dim}(X)$, such that every cohomology class $x\in H^j(X;\mathbb Z_2)$, $0\leq j\leq k$, is a polynomial in the Stiefel-Whitney classes $w_i(\xi)$. In this note we compute the characteristic rank of vector bundles over the Stiefel manifold $V_k(\mathbb F^n)$, $\mathbb F=\mathbb R,\mathbb C,\mathbb H$

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The mod 2 cohomology rings of oriented Grassmannians via Koszul complexes

Matszangosz,

Wendt

2024

Math. Z.

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We study the structure of mod 2 cohomology rings of oriented Grassmannians $$\widetilde{{\text {Gr}}}_k(n)$$ Gr ~ k ( n ) of oriented k-planes in $${\mathbb {R}}^n$$ R n . Our main focus is on the structure of the cohomology ring $$\textrm{H}^*(\widetilde{{\text {Gr}}}_k(n);{\mathbb {F}}_2)$$ H ∗ ( Gr ~ k ( n ) ; F 2 ) as a module over the characteristic subring C, which is the subring generated by the Stiefel–Whitney classes $$w_2,\ldots ,w_k$$ w 2 , … , w k . We identify this module structure using Koszul complexes, which involves the syzygies between the relations defining C. We give an infinite family of such syzygies, which results in a new upper bound on the characteristic rank of $$\widetilde{{\text {Gr}}}_k(2^t)$$ Gr ~ k ( 2 t ) , $$k<2^t$$ k < 2 t , and formulate a conjecture on the exact value of the characteristic rank of $$\widetilde{{\text {Gr}}}_k(n)$$ Gr ~ k ( n ) . For the case $$k=3$$ k = 3 , we use the Koszul complex to compute a presentation of the cohomology ring $$H=\textrm{H}^*(\widetilde{{\text {Gr}}}_3(n);{\mathbb {F}}_2)$$ H = H ∗ ( Gr ~ 3 ( n ) ; F 2 ) for $$2^{t-1}<n\le 2^t-4$$ 2 t - 1 < n ≤ 2 t - 4 for $$t\ge 4$$ t ≥ 4 , complementing existing descriptions in the cases $$n=2^t-i$$ n = 2 t - i , $$i=0,1,2,3$$ i = 0 , 1 , 2 , 3 for $$t\ge 3$$ t ≥ 3 . More precisely, as a C-module, H splits as a direct sum of the characteristic subring C and the anomalous module H/C, and we compute a complete presentation of H/C as a C-module from the Koszul complex. We also discuss various issues that arise for the cases $$k>3$$ k > 3 , supported by computer calculation.

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The cup-length of the oriented Grassmannians vs a new bound for zero-cobordant manifolds

Cited by 19 publications

References 3 publications

On the cohomology ring and upper characteristic rank of Grassmannian of oriented 3-planes

On the cohomology ring and upper characteristic rank of Grassmannian of oriented 3-planes

Characteristic rank of vector bundles over Stiefel manifolds

The mod 2 cohomology rings of oriented Grassmannians via Koszul complexes

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