arXiv subjects
Publications and source records attributed to Bikramjit Kundu.
In this note, we compute the upper characteristic rank of the projective Stiefel manifolds over $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$ and the flip Stiefel manifolds. We also provide bounds for the $\mathrm{cup}$ lengths of these spaces. We also provide necessary conditions for the existence of $S^3$-map between quaternionic Stiefel manifolds using the Fadell-Husseini index.
This article explores equivariant localization in the category of $G$-spaces, where $G$ is a compact Lie group. We establish a commutation rule for the localization functor and the equivariant loop functor. Additionally, we introduce and classify certain good objects in this category up to their Bredon cohomology with coefficients in the constant rational Mackey functor $\underline{\Q}$.
This paper computes the Fadell-Husseini index of Stiefel manifolds in the case where the group acts via permutations of the orthogonal vectors. The computations are carried out in the case of elementary Abelian $p$-groups. The results are shown to imply certain generalizations of the Kakutani-Yamabe-Yujobo theorem.
A space $X$ is $W$-trivial if for every real vector bundle $α$ over $X$ the total Stiefel-Whitney class $w(α)$ is 1. It follows from a result of Milnor that if $X$ is an orientable closed smooth manifold of dimension $1,2,4$ or $8$, then $X$ is not $W$-trivial. In this note we completely characterize $W$-trivial orientable connected closed smooth manifolds in dimensions $3,5$ and $6$. In dimension $7$, we describe necessary conditions for an orientable connected closed smooth $7$-manifold to be $W$-trivial.
In this paper, we consider the flag manifold of $p$ orthogonal subspaces of equal dimension which carries an action of the cyclic group of order $p$. We provide a complete calculation of the associated Fadell-Husseini index. This may be thought of as an odd primary version of the computations of Baralić et al [Forum Math., 30 (2018), pp. 1539--1572] for the Grassmann manifold $G_n(\mathbb{R}^{2n})$. These results have geometric consequences for $p$-fold orthogonal shadows of a convex body.
In this paper, we consider numerical indices associated to spaces with free $C_p$-action. We prove that the Stiefel manifolds provide an example of non-tidy spaces for $p=2$, which are those whose co-index and index disagree. In the case of odd primes, we construct examples of co-index $3$ whose index may be arbitrarily large.