Finiteness of Total Cofibres

Hüttemann, Thomas

Department of Mathematical Sciences, University of Aberdeen, Meston Building, King's College, Aberdeen AB24 3UE, UK. e-mail: [email protected]

K-Theory 31(2):p 101-123, February 2004.

The suspension of the total cofibre Γ(Y) of a diagram Y of infinite CW complexes, indexed by the face lattice of a polytope, is in general not homotopy equivalent to a finite CW complex. However, if Y satisfies additional conditions (making it into a locally finite non-linear sheaf on a projective toric variety) we can show that ΣΓ(Y) is a homotopy finite space. The main tools used in the proof are a Bousfield–Kan spectral sequence for total cofibres, and a modification of the Čech construction to calculate sheaf cohomology groups.