Making Lifting Obstructions Explicit

Original · EN

If P → X is a topological principal K-bundle and K a central extension of K by Z, then there is a natural obstruction class δ₁(P) in H²(X, Z) in sheaf cohomology whose vanishing is equivalent to the existence of a K-bundle P over X with P P/Z. In this paper we establish a link between homotopy theoretic data and the obstruction class δ₁(P) which in many cases can be used to calculate this class in explicit terms. Writing ∂dᵖ πd(X) → πd₋₁(K) for the connecting maps in the long exact homotopy sequence, two of our main results can be formulated as follows. If Z is a quotient of a contractible group by the discrete group Γ, then the homomorphism π₃(X) → Γinduced by δ₁(P) ∈ H²(X, Z) H³ sing(X,Γ) coincides with ∂₂ K ∘ ∂₃ᵖ and if Z is discrete, then δ₁(P) ∈ H²(X, Z) induces the homomorphism -∂₁ K ∘ ∂₂ᵖ π₂(X) → Z. We also obtain some information on obstruction classes defining trivial homomorphisms on homotopy groups.