On sectioning multiples of the nontrivial line bundle over Grassmannians

Horanská, Ľubomíra. "On sectioning multiples of the nontrivial line bundle over Grassmannians." Proceedings of the 17th Winter School "Geometry and Physics". Palermo: Circolo Matematico di Palermo, 1998. [59]-64. <http://eudml.org/doc/221449>.

@inProceedings{Horanská1998,
abstract = {Let $G_\{n,k\}$ ($\widetilde\{G\}_\{n,k\}$) denote the Grassmann manifold of linear $k$-spaces (resp. oriented $k$-spaces) in $\{\mathbb \{R\}\}^\{n\}$, $d_\{n,k\} = k(n-k) = \text\{dim\} G_\{n,k\}$ and suppose $n \ge 2k$. As an easy consequence of the Steenrod obstruction theory, one sees that $(d_\{n,k\}+1)$-fold Whitney sum $(d_\{n,k\}+1)\xi _\{n,k\}$ of the nontrivial line bundle $\xi _\{n,k\}$ over $G_\{n,k\}$ always has a nowhere vanishing section. The author deals with the following question: What is the least $s$ ($= s_\{n,k\}$) such that the vector bundle $s\xi _\{n,k\}$ admits a nowhere vanishing section ? Obviously, $s_\{n,k\} \le d_\{n,k\}+1$, and for the special case in which $k=1$, it is known that $s_\{n,1\} = d_\{n,1\}+1$. Using results of J. Korbaš and P. Sankaran [Proc. Indian Acad. Sci., Math. Sci. 101, No. 2, 111-120 (1991; Zbl 0745.55003)], S. Gitler and D. Handel [Topology 7, 39-46 (1968; Zbl 0166.19405)] and the Dai-Lam level of $\widetilde\{G\}_\{n,k\}$ with !},
author = {Horanská, Ľubomíra},
booktitle = {Proceedings of the 17th Winter School "Geometry and Physics"},
keywords = {Proceedings; Winter school; Geometry; Physics; Srní (Czech Republic)},
location = {Palermo},
pages = {[59]-64},
publisher = {Circolo Matematico di Palermo},
title = {On sectioning multiples of the nontrivial line bundle over Grassmannians},
url = {http://eudml.org/doc/221449},
year = {1998},
}

TY - CLSWK
AU - Horanská, Ľubomíra
TI - On sectioning multiples of the nontrivial line bundle over Grassmannians
T2 - Proceedings of the 17th Winter School "Geometry and Physics"
PY - 1998
CY - Palermo
PB - Circolo Matematico di Palermo
SP - [59]
EP - 64
AB - Let $G_{n,k}$ ($\widetilde{G}_{n,k}$) denote the Grassmann manifold of linear $k$-spaces (resp. oriented $k$-spaces) in ${\mathbb {R}}^{n}$, $d_{n,k} = k(n-k) = \text{dim} G_{n,k}$ and suppose $n \ge 2k$. As an easy consequence of the Steenrod obstruction theory, one sees that $(d_{n,k}+1)$-fold Whitney sum $(d_{n,k}+1)\xi _{n,k}$ of the nontrivial line bundle $\xi _{n,k}$ over $G_{n,k}$ always has a nowhere vanishing section. The author deals with the following question: What is the least $s$ ($= s_{n,k}$) such that the vector bundle $s\xi _{n,k}$ admits a nowhere vanishing section ? Obviously, $s_{n,k} \le d_{n,k}+1$, and for the special case in which $k=1$, it is known that $s_{n,1} = d_{n,1}+1$. Using results of J. Korbaš and P. Sankaran [Proc. Indian Acad. Sci., Math. Sci. 101, No. 2, 111-120 (1991; Zbl 0745.55003)], S. Gitler and D. Handel [Topology 7, 39-46 (1968; Zbl 0166.19405)] and the Dai-Lam level of $\widetilde{G}_{n,k}$ with !
KW - Proceedings; Winter school; Geometry; Physics; Srní (Czech Republic)
UR - http://eudml.org/doc/221449
ER -