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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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“…We note that Theorem A (cf. Theorem 3.4 in this article) gives a different proof of Theorem 1.1 of [11].…”

Section: Theorem a (A)mentioning

confidence: 85%

“…Due to [4] and [11], we know the degree of the first indecomposable element of H * ( G n,3 ; Z 2 ) after degree 3. In this article we determine the complete set of degrees of indecomposables in H * ( G n,3 ; Z 2 ) (cf.…”

Section: Introductionmentioning

confidence: 99%

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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“…Although the characteristic rank of γ n,3 is already known [10], together with the corresponding result on cup-length, we will explicitly derive the values implied by Theorem 4.1 in this case. The intent is to allow us to discuss certain aspects of the proof, which generalize to the case of arbitrary k. Proof.…”

Section: The Resultsmentioning

confidence: 99%

“…In particular, the paper by Fukaya [2] (where a slightly different notation for Grassmann manifolds is used; G n,3 corresponds to G n+3,3 in this paper) contains the proof, that cup( G 2 t −1,3 ) = 2 t − 3 for t ≥ 3, and the following interesting conjecture [ The value cup( G 2 t −1,3 ) = 2 t − 3 has been obtained independently by Korbaš [4] employing an approach using the notion of characteristic rank. Making use of refined version of this idea, some other parts of the conjecture have been proved in papers [10], [11]. Namely, the cases corresponding to n in the interval 2 t −1 ≤ n < 2 t −1+ 2 t 3 for t ≥ 3 and n = 2 t + 2 t−1 + a for a = 1, 2 and t ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

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Bounds for the characteristic rank and cup-length of oriented Grassmann manifolds

Rusin¹

2018

Arch. Math. (Brno)

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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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Characteristic rank of canonical vector bundles over oriented Grassmann manifolds G˜3,n

Cited by 5 publications

References 10 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

Bounds for the characteristic rank and cup-length of oriented Grassmann manifolds

The mod 2 cohomology rings of oriented Grassmannians via Koszul complexes

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