Instanton Floer Homology and Binary Polyhedral Spaces

To a mathematician, a surface is a shape that, upon close enough inspection, looks like a flat piece of paper around each of its points. Thus, to an ant, the surface of a bagel is conceived as flat, while the (roughly) spherical surface of the earth looks flat to us. Nevertheless, on a global level these shapes are topologically distinct in the sense that one cannot deform one into the other through stretching, bending and shrinking. Manifolds generalize the notion of a surface to other dimensions. In this terminology, a surface is a 2-manifold and a curve is a 1-manifold. These low-dimensional shapes have been completely classified. Much progress has been made in dimension 3, but several serious questions still remain. In my thesis, I have explored a recent version of an algebraic invariant known as instanton Floer homology associated with a restricted class of 3-manifolds. It is built from geometric data extracted from the solution spaces of the instanton equation -- a partial differential equation special to dimension 3 and 4. I have developed algebra needed to properly define this invariant and provided complete calculations for the binary polyhedral spaces -- a family of 3-manifolds intimately linked with the platonic solids and their symmetries. I have also employed techniques from quiver theory to construct several hyper-Kähler bordisms between members of this family