Kawasaki has calculated the integral homology groups of weighted projective space, and his results imply the existence of a homotopy equivalence between weighted projective space and a CW-complex, with a single cell in each even dimension less than or equal to that of weighted projective space. When the weights satisfy certain divisibility conditions then the associated weighted projective space is actually homeomorphic to such an minimal CW-complex and such decompositions are well-known in these cases. Otherwise this minimal CW-complex is not a weighted projective space. Our aim is to give an explicit CW-structure on any weighted projective space, using an invariant decomposition of complex projective space with respect to the action of a product of finite cyclic groups. The result has many cells, in both odd and even dimensions; nevertheless, we identify it with a subdivision of the minimal decomposition whenever the weights are divisive. We then extend the decomposition to stunted weighted projective space, defined as the quotient of one weighted projective space by another. Finally, we compute the integral homology groups of stunted weighted projective space, identify generators in terms of cellular cycles, and describe cup products in the corresponding cohomology ring.

K3 surfaces have garnered attention across various fields, from optics and dynamics to high energy physics, making them a subject of extensive study for many decades. Recent work by mathematicians, including Brandhorst [1], has focused on non-symplectic automorphisms, aiming to categorize K3 surfaces that admit such automorphisms. Brandhorst made a list of unique K3 surfaces with purely non-symplectic automorphisms and established specific criteria for a K3 surface to be isomorphic to one on his list. This thesis aims to provide an alternative representation of select K3 surfaces from Brandhorst's list. While Brandhorst predominantly characterizes these surfaces as elliptic K3 surfaces, we offer a description of these surfaces as hypersurfaces in weighted projective space. Our approach involves verifying the criteria established by Brandhorst, thereby establishing an isomorphism between the surfaces in question. Through this study, we contribute to the understanding of K3 surfaces and their automorphisms while also demonstrating the correspondence between different spaces and methodologies for analyzing K3 surfaces. This work lays the groundwork for further investigations into K3 surfaces with purely non-symplectic automorphisms, paving the way for deeper insights into their structural properties and geometric intricacies.