Aleksandar Vučić | Searcharxiv

arXiv2022

Universal complexes in toric topology

We study combinatorial and topological properties of the universal complexes $X(\mathbb{F}_p^n)$ and $K(\mathbb{F}_p^n)$ whose simplices are certain unimodular subsets of $\mathbb{F}_p^n$. We calculate their $\mathbf f$-vectors and their Tor-algebras, show that they are shellable but not shifted, and find their applications in toric topology and number theory. We showed that the Lusternick-Schnirelmann category of the moment angle complex of $X(\mathbb{F}_p^n)$ is $n$, provided $p$ is an odd prime, and the Lusternick-Schnirelmann category of the moment angle complex of $K(\mathbb{F}_p^n)$ is $[\frac n 2]$. Based on the universal complexes, we introduce the Buchstaber invariant $s_p$ for a prime number $p$.

math.GT↗

arXiv2018

Orthogonal shadows and index of Grassmann manifolds

In this paper we study the $\Z/2$ action on real Grassmann manifolds $G_{n}(\R^{2n})$ and $\widetilde{G}_{n}(\R^{2n})$ given by taking (appropriately oriented) orthogonal complement. We completely evaluate the related $\Z/2$ Fadell--Husseini index utilizing a novel computation of the Stiefel--Whitney classes of the wreath product of a vector bundle. These results are used to establish the following geometric result about the orthogonal shadows of a convex body: For $n=2^a (2 b+1)$, $k=2^{a+1}-1$, $C$ a convex body in $\R^{2n}$, and $k$ real valued functions $α_1,\ldots,α_k$ continuous on convex bodies in $\R^{2n}$ with respect to the Hausdorff metric, there exists a subspace $V\subseteq\R^{2n}$ such that projections of $C$ to $V$ and its orthogonal complement $V^{\perp}$ have the same value with respect to each function $α_i$, which is $α_i (p_V(C))=α_i (p_{V^\perp} (C))$ for all $1\leq i\leq k$.

math.AT↗