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, which we define explicitly in the paper.

Author Bio

Shamuel Auyeung is currently studying mathematics and philosophy. His near-future plans include studying mathematics and doing research at the graduate level. He enjoys cooking, riding his bike, flying kites, and talking about the philosophy of Soren Kierkegaard.

Joshua Ruiter graduated from Calvin College in spring of 2016 and began the mathematics Ph.D. program at Michigan State University the following fall. He also enjoys playing the cello and arranging pop music for cello quartet.

Daiwei Zhang grew up in Beijing and came to the U. S. when he was 16. After finishing high school in Colorado, he went to Calvin College and graduated with a B. S. in mathematics. He is now pursuing a Ph.D. in biostatistics at the University of Michigan. He enjoys learning mathematics because it shows him the greatness of God, and statistics because it shows him the finiteness of man.