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.
James M. Turner, Department of Mathematics, Calvin College
Auyeung, Shamuel; Ruiter, Joshua; and Zhang, Daiwei
"An Algebraic Characterization of Highly Connected 2n-Manifolds,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 17
, Article 5.
Available at: https://scholar.rose-hulman.edu/rhumj/vol17/iss2/5