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The index of equidimensional flag manifolds

Samik Basu, Bikramjit Kundu

TL;DR

This work computes the Fadell–Husseini index for the equidimensional flag manifolds $F_n(k)$ under a free $C_p$ action, forming an odd-primary analogue of Grassmannian results and linking to geometric shadow problems for convex bodies. The authors develop the cohomology of equidimensional flag manifolds, introduce the $p$-fold wreath power of spaces and vector bundles, and derive explicit formulas for the associated Chern and Pontrjagin classes. They reduce index computations to universal wreath-power characteristic classes and use Serre spectral sequences to obtain exact index ideals, obtaining closed forms like ${\rm Index}_{C_p}(F_n({\mathbb C}))=(uv^{p^{a+1}-1},v^{p^{a+1}})$ when $n=p^a q$ with $p\nmid q$, with corresponding real-case analogues. The results provide precise obstructions to equivariant maps and have implications for $p$-fold orthogonal shadows of convex bodies, illustrating a powerful link between wreath-power topology and geometric combinatorics.

Abstract

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.

The index of equidimensional flag manifolds

TL;DR

This work computes the Fadell–Husseini index for the equidimensional flag manifolds under a free action, forming an odd-primary analogue of Grassmannian results and linking to geometric shadow problems for convex bodies. The authors develop the cohomology of equidimensional flag manifolds, introduce the -fold wreath power of spaces and vector bundles, and derive explicit formulas for the associated Chern and Pontrjagin classes. They reduce index computations to universal wreath-power characteristic classes and use Serre spectral sequences to obtain exact index ideals, obtaining closed forms like when with , with corresponding real-case analogues. The results provide precise obstructions to equivariant maps and have implications for -fold orthogonal shadows of convex bodies, illustrating a powerful link between wreath-power topology and geometric combinatorics.

Abstract

In this paper, we consider the flag manifold of orthogonal subspaces of equal dimension which carries an action of the cyclic group of order . 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 . These results have geometric consequences for -fold orthogonal shadows of a convex body.
Paper Structure (5 sections, 28 theorems, 166 equations)

This paper contains 5 sections, 28 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Cohomology of flag manifolds
  3. Cohomology of the wreath power of spaces
  4. The wreath power of vector bundles
  5. Index computations

Key Result

Theorem 1.1

a) Let $n=p^aq$ such that $p\nmid q$, and $r\leq 2(\frac{p^{a+1}-1}{p-1})$. Let $\alpha_1,\cdots, \alpha_r : \hbox{Convex} ({\mathbb C}^{pn})\to {\mathbb R}$. For every proper convex body $C\subset {\mathbb C}^{pn}$, there exist $p$ mutually orthogonal $n$-dimensional subspaces $V_1,\cdots, V_p$ of b) Let $n$, $p$, $a$ and $q$ as above, and $r< 2(\frac{p^{a+1}-1}{p-1})$. Let $\alpha_1,\cdots, \al

Theorems & Definitions (48)

  • Theorem 1.1
  • proof
  • Corollary 1.2
  • Theorem 1.3
  • proof
  • Proposition 2.3
  • proof
  • Theorem 2.5
  • proof
  • Proposition 2.6
  • ...and 38 more