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.

Thesis advisor: Joshua E. Greene / We develop and implement obstructions to realizing a 3-manifold all of whose prime summands are lens spaces as Dehn surgery on a knot K in the Poincaré homology sphere, and in the process, we determine the knot Floer homology groups of a knot with such a surgery. We show that such a surgery never results in a 3-manifold with more than three non-trivial summands, and that if the result of surgery has exactly three non-trivial summands, then K is isotopic to a regular Seifert fiber. We furthermore identify the only two knots with half-integer lens space surgeries, and thus complete the classification of knots in the Poincaré homology sphere with non-integer lens space surgeries. We lastly show that a lens space L(p, q) that is realized as integer surgery on a knot K is realized as integer surgery on a Tange knot when p ≥ 2g(K). In order to do so, we build on Greene’s work on changemaker lattices and develop the theory of E8-changemaker lattices. / Thesis (PhD) — Boston College, 2023. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.