Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

LS-category and topological complexity of Milnor manifolds and corresponding generalized projective product spaces

Navnath Daundkar

Abstract

Milnor manifolds are a class of certain codimension-$1$ submanifolds of the product of projective spaces. In this paper, we study the LS-category and topological complexity of these manifolds. We determine the exact value of the LS-category and in many cases, the topological complexity of these manifolds. We also obtain tight bounds on the topological complexity of these manifolds. It is known that Milnor manifolds admit $\mathbb{Z}_2$ and circle actions. We compute bounds on the equivariant LS-category and equivariant topological complexity of these manifolds. Finally, we describe the mod-$2$ cohomology rings of some generalized projective product spaces corresponding to Milnor manifolds and use this information to compute the bound on LS-category and topological complexity of these spaces.

LS-category and topological complexity of Milnor manifolds and corresponding generalized projective product spaces

Abstract

Milnor manifolds are a class of certain codimension- submanifolds of the product of projective spaces. In this paper, we study the LS-category and topological complexity of these manifolds. We determine the exact value of the LS-category and in many cases, the topological complexity of these manifolds. We also obtain tight bounds on the topological complexity of these manifolds. It is known that Milnor manifolds admit and circle actions. We compute bounds on the equivariant LS-category and equivariant topological complexity of these manifolds. Finally, we describe the mod- cohomology rings of some generalized projective product spaces corresponding to Milnor manifolds and use this information to compute the bound on LS-category and topological complexity of these spaces.
Paper Structure (8 sections, 39 theorems, 89 equations)

This paper contains 8 sections, 39 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. LS-category, topological complexity and Milnor manifolds
  3. LS-category and topological complexity
  4. Milnor manifolds
  5. LS-category and topological complexity of Milnor manifolds
  6. Equivariant LS-category and equivariant topological complexity of Milnor manifolds
  7. Generalized projective product spaces with one of the components as Milnor manifolds
  8. Acknowledgement

Key Result

Theorem 2.3

If $X$ is a path connected, paracompact topological space. Then

Theorems & Definitions (71)

  • Definition 2.1: colmangranteqtc
  • Definition 2.2
  • Theorem 2.3: FarberTC
  • Theorem 2.4: FarberTC
  • Theorem 2.5: Fox
  • Theorem 2.6: CLOT
  • Theorem 2.7: CLOT
  • Theorem 2.8: FarberTC
  • Theorem 2.9: Mukherjee
  • Theorem 2.10: Mukherjee
  • ...and 61 more