Weighted projective spaces and iterated Thom spaces

Abstract

For any weight vector $\chi$ of positive integers, the weighted projective space $\mathbb{P}(\chi)$ is a projective toric variety, and has orbifold singularities in every case other than standard projective space. Our principal aim is to study the algebraic topology of $\mathbb{P}(\chi)$, paying particular attention to its localisation at individual primes $p$. We identify certain $p$-primary weight vectors $\pi$ for which $\mathbb{P}(\pi)$ is homeomorphic to an iterated Thom space, and discuss how any weighted projective space may be reassembled from its $p$-primary parts. The resulting Thom isomorphisms provide an alternative to Kawasaki's calculation of the cohomology ring of $\mathbb{P}(\chi)$, and allow us to recover Al Amrani's extension to complex $K$-theory. Our methods generalise to arbitrary complex oriented cohomology algebras and their dual homology coalgebras, as we demonstrate for complex cobordism theory, the universal example. In particular, we describe a fundamental class that belongs to the complex bordism coalgebra of $\mathbb{P}(\chi)$, and may be interpreted as a resolution of singularities.