UCLA

We show that, for a complete simplicial toric variety X, we can determine its homotopy K-theory (denoted KH-theory) entirely in terms of the torus pieces of open sets forming an open cover of X. We accomplish this by constructing a simplicial scheme BOT_X and constructing a relationship between the spectrum KH(X) and a certain spectrum determined by BOT_X. Using our construction of BOT_X, we construct conditions under which, given two complete simplicial toric varieties with the same simplicial structure, we can induce a morphism from BOT_X to BOT_Y that is, in each degree, component-wise an isogeny. This allows us to show that, under these conditions, the two spectra KH(X) ⊗ Q and KH(Y) ⊗ Q are weakly equivalent. We then apply this result to determine the rational KH-theory of weighted projective spaces. We next turn our attention to calculating the F_K groups for complete toric surfaces and 2-dimensional weighted projective spaces. This allows us to determine K_n(P(a,b,c)) ⊗ Q for n ≤ 0, and allows us to conclude that complete toric surfaces and 2-dimensional weighted projective spaces are K₀-regular. We conclude by determining conditions under which our approach for dimension 2 works in arbitrary dimensions.