Making Lifting Obstructions Explicit
Résumé
If P \to X is a topological principal K-bundle and \hat K a central extension of K by Z, then there is a natural obstruction class \delta_1(P) in \check H^2(X,\uline Z) in sheaf cohomology whose vanishing is equivalent to the existence of a \hat K-bundle \hat P over X with P \cong \hat P/Z. In this paper we establish a link between homotopy theoretic data and the obstruction class \delta_1(P) which in many cases can be used to calculate this class in explicit terms. Writing \partial_d^P \: \pi_d(X) \to \pi_{d-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 \Gamma, then the homomorphism \pi_3(X) \to \Gamma induced by \delta_1(P) \in \check H^2(X,\uline Z) \cong H^3_{\rm sing}(X,\Gamma) coincides with \partial_2^{\hat K} \circ \partial_3^P and if Z is discrete, then \delta_1(P) \in \check H^2(X,\uline Z) induces the homomorphism -\partial_1^{\hat K} \circ \partial_2^P \: \pi_2(X) \to Z. We also obtain some information on obstruction classes defining trivial homomorphisms on homotopy groups.