## Abstract

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*,2*n*)-cell complex: forming a space by attaching a (2*n*-1)-sphere into a wedge sum of *n*-spheres. In this paper, we classify oriented, (*n*-1)-connected, compact and closed 2*n*-manifolds up to homotopy by treating them as (*n*,2*n*)-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*,2*n*)-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.

## Faculty Sponsor

James M. Turner

## Recommended Citation

Auyeung, Shamuel; Ruiter, Joshua; and Zhang, Daiwei
(2016)
"An Algebraic Characterization of Highly Connected 2π-Manifolds,"
*Rose-Hulman Undergraduate Mathematics Journal*: Vol. 17:
Iss.
2, Article 5.

Available at:
https://scholar.rose-hulman.edu/rhumj/vol17/iss2/5