| Title:
|
A note on the cohomology ring of the oriented Grassmann manifolds $\widetilde{G}_{n,4}$ (English) |
| Author:
|
Rusin, Tomáš |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
55 |
| Issue:
|
5 |
| Year:
|
2019 |
| Pages:
|
319-331 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We use known results on the characteristic rank of the canonical $4$–plane bundle over the oriented Grassmann manifold $\widetilde{G}_{n,4}$ to compute the generators of the $\mathbb{Z}_2$–cohomology groups $H^j(\widetilde{G}_{n,4})$ for $n=8,9,10,11$. Drawing from the similarities of these examples with the general description of the cohomology rings of $\widetilde{G}_{n,3}$ we conjecture some predictions. (English) |
| Keyword:
|
oriented Grassmann manifold |
| Keyword:
|
characteristic rank |
| Keyword:
|
Stiefel-Whitney class |
| MSC:
|
55R25 |
| MSC:
|
57R20 |
| MSC:
|
57T15 |
| idZBL:
|
Zbl 07144746 |
| idMR:
|
MR4057928 |
| DOI:
|
10.5817/AM2019-5-319 |
| . |
| Date available:
|
2019-12-09T12:24:06Z |
| Last updated:
|
2020-11-24 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147944 |
| . |
| Reference:
|
[1] Basu, S., Chakraborty, P.: On the cohomology ring and upper characteristic rank of Grassmannian of oriented 3-planes.arXiv:1712.00284v1 [math.AT]. |
| Reference:
|
[2] Borel, A.: La cohomologie mod $2$ de certains espaces homogènes.Comment. Math. Helv. 27 (1953), 165–197. MR 0057541, 10.1007/BF02564561 |
| Reference:
|
[3] Korbaš, J.: The cup-length of the oriented Grassmannians vs a new bound for zero-cobordant manifolds.Bull. Belg. Math. Soc. Simon Stevin 17 (2010), 69–81. MR 2656672, 10.36045/bbms/1267798499 |
| Reference:
|
[4] Korbaš, J., Rusin, T.: A note on the $\mathbb{Z}_2$-cohomology algebra of oriented Grassmann manifolds.Rend. Circ. Mat. Palermo, II. Ser. 65 (2016), 507–517. MR 3571326 |
| Reference:
|
[5] Korbaš, J., Rusin, T.: On the cohomology of oriented Grassmann manifolds.Homology Homotopy Appl. 18 (2) (2016), 71–84. MR 3518983, 10.4310/HHA.2016.v18.n2.a4 |
| Reference:
|
[6] Milnor, J., Stasheff, J.: Characteristic Classes.Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. Zbl 0298.57008, MR 0440554 |
| Reference:
|
[7] Naolekar, A.C., Thakur, A.S.: Note on the characteristic rank of vector bundles.Math. Slovaca 64 (2014), 1525–1540. MR 3298036, 10.2478/s12175-014-0289-4 |
| Reference:
|
[8] Prvulović, B.I., Radovanović, M.: On the characteristic rank of vector bundles over oriented Grassmannians.Fundamenta Mathematicae 244 (2019), 167–190. MR 3874670, 10.4064/fm470-3-2018 |
| Reference:
|
[9] Stong, R.E.: Semicharacteristics of oriented Grassmannians.J. Pure Appl. Alg. 33 (1984), 97–103. MR 0750235, 10.1016/0022-4049(84)90029-X |
| . |
