Making Lifting Obstructions Explicit

TL;DR

This paper links homotopy theory with sheaf cohomology to explicitly compute obstructions to lifting principal bundles through central extensions, providing new formulas involving homotopy groups.

Contribution

It establishes explicit relations between homotopy data and obstruction classes for lifting principal K-bundles, enabling concrete calculations in various cases.

Findings

Homomorphism ext{π}_3(X) o ext{Γ} matches a composition of connecting maps.

Obstruction class ext{δ}_1(P) induces specific homomorphisms on ext{π}_2(X).

Results apply when Z is a quotient of a contractible group or discrete.

Abstract

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.