Topological complexity of oriented Grassmann manifolds
Uroš A. Colović, Branislav I. Prvulović, Marko Radovanović
TL;DR
This work computes the $\mathbb Z_2$-zero-divisor cup-length of oriented Grassmannians $\widetilde G_{n,3}$ by leveraging the twofold covering to a real Grassmannian, along with a Gröbner-basis description of the subalgebra $W_n$ generated by $\widetilde w_2,\widetilde w_3$. By deriving explicit $\operatorname{zcl}(W_n)$ values across many $n$-intervals and establishing monotonicity, the authors obtain sharp lower bounds for $\operatorname{TC}(\widetilde G_{n,3})$ in substantial ranges, with exact results for infinitely many $n$. They show $\operatorname{zcl}(\widetilde G_{n,3})\ge1+\operatorname{zcl}(W_n)$ for all $n\ge6$, and provide upper bounds $\operatorname{zcl}(\widetilde G_{n,3})\le2+\operatorname{zcl}(W_n)$ for large $n$, supported by extensive computations up to $n=100$ and a strong conjecture that equality holds universally. The results illuminate the relationship between the algebraic structure of $W_n$ and the topological complexity of oriented Grassmannians.
Abstract
We study the $\mathbb Z_2$-zero-divisor cup-length, denoted by $\operatorname{zcl}_{\mathbb Z_2}(\widetilde G_{n,3})$, of the Grassmann manifolds $\widetilde G_{n,3}$ of oriented $3$-dimensional vector subspaces in $\mathbb R^n$. Some lower and upper bounds for this invariant are obtained for all integers $n\ge6$. For infinitely many of them the exact value of $\operatorname{zcl}_{\mathbb Z_2}(\widetilde G_{n,3})$ is computed, and in the rest of the cases these bounds differ by 1. We thus establish lower bounds for the topological complexity of Grassmannians $\widetilde G_{n,3}$.