NAVNATH DAUNDKAR A. 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 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.

A. In this article, we introduce the notion of a wedge of graphs and provide detailed computations for the independence complex of a wedge of path and cycle graphs. In particular, we show that these complexes are either contractible or wedges of spheres.

The moduli space of chains in the plane with generic side lengths is a smooth, closed manifold. Interestingly, this manifold is also the fixed point set of the complex conjugation on a toric variety, known as the abelian polygon space. Hence, affording the structure of a real toric variety. In this paper we show that the moment polytope of the moduli space of chains can be determined by the combinatorial data, called the short code of the length vector. We also classify chain spaces which are aspherical using a result of Davis, Januszkiewicz and Scott.