# The Cup-Length of Stiefel and Projective Stiefel Manifolds

Július Korbaš, Renate Scheidler, Peter Zvengrowski – GJM, Volume 6, Issue 1 (2021), 27-34.

This paper discusses some generalities about cup-length of manifolds and then gives an explicit formula for the $\mathbb Z_2$-cup-length of the Stiefel manifolds $V_{n,r}$, as well as strong lower bounds for the $\mathbb Z_2$-cup-length of the projective Stiefel manifolds $X_{n,r}$, for all $1 \leq r\leq n-1$. A simple formula relating the two cases is given.

We also show the consequences for the Lyusternik-Shnirel’man category, as well as a family of interesting number theoretical identities that arise from the $V_{n,r}$ calculations.

Categories: Issue1

#### Milestones:

Accepted: September 5, 2020
Revised: March 25, 2021
Published online: May 29, 2021

#### Authors:

Július Korbaš
Department of Algebra and Geometry,
Faculty of Mathematics, Physics, and Informatics,
Comenius University, Mlynska dolina,
SK-842 48 Bratislava 4, Slovakia.

Renate Scheidler
Department of Mathematics and Statistics,
The University of Calgary, Calgary,