Topological complexity and Lusternik Schnirelmann category of manifolds

Lusternik schnirelmann category and topological complexity are important invariant of topological spaces, now a days a lot of mathematician are interested to work in this area. In this paper in order to detect properties of spaces, we will compute Lusternik schnirelmann category and topological complexity of some of these spaces by computing the cup-length and zero-cup-length. These include the manifolds that we will calculate for the topological complexity and Lusternik Schnirelmann category are some of the Gressmannian manifolds, such as G_2 (R^4), and the products of manifolds, especially the products of the real projective spaces and their wedge products. Let TC(X) denotes the topological complexity of the path connected topological space X, and also cat(X) denots the Lusternik Schnirelmann category of topological space X. In the calculation of these numbers, we will first compute the upper and lower bounds of these invariants for considerable spaces, and we will try to approximate the boundaries with the methods and techniques to get the exact number. In this paper, we will use the cup-length and zero divisors cup length of spaces as the lower bounds for calculating TC and cat that are important computational tools for calculating these numbers.