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Cellular structures and stunted weighted projective space

O'Neill, Beverley

[Thesis]. Manchester, UK: The University of Manchester; 2014.

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Abstract

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.