An Algebraic Characterization of Highly Connected $2n$-Manifolds

All surfaces, up to homeomorphism, can be formed by gluing the edges of a polygon. This process is generalized into the idea of a $(n, 2n)$-cell complex: forming a space by attaching a $(2n−1)$-sphere into a wedge sum of n-spheres. In this paper, we classify oriented, $(n-1)$-connected, compact and closed $2n$-manifolds up to homotopy by treating them as $(n, 2n)$-cell complexes. To simplify the calculation, we create a basis called the Hilton basis for the homotopy class of the attaching map of the $(n, 2n)$-cell complex. At the end, we show that two attaching maps give the same, up to homotopy, manifold if and only if their homotopy classes, when written in a Hilton basis, differ only by a change-of-basis matrix that is in the image of a certain map $\Phi$, which we define explicitly in the paper.