In this project, we studied the Morse Theory and use it to prove Reeb's theorem, which states that if If $M$ is a compact manifold and $f$ is a $C^2$-function on $M$ with only two critical points, both of which are non-degenerate, then $M$ is homeomorphic to a sphere. We followed the book Morse Theory by John Milnor.

Complex Grassmannian $G_k(\mathbb{C}^n)$ consists of all $k$ dimensional complex subspace in $\mathbb{C}^n$. It is a smooth (complex) manifold of real dimension $2k(n-k)$. Using Morse theory, we studied the cell structure of complex Grassmannian by constructing an explicit Morse function on it.

Here, we again studied the cell structure of complex Grassmannian topologically i.e., without using any smooth structure of complex grassmannian. This process is equally applicable for real Grassmannian too. Finally, we compare this cell structure of complex Grassmannian with that of obtained using Morse theory in project II.

Using the CW-structure of the complex Grassmannian, we first compute the cellular homology of it and then cohomology by Poincaré duality. Finally, compute the cohomology ring structute of it using Young Tableau.