# Timelike homotopy groups and a characteristic class for lorentz manifolds

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## Abstract

Let M be a manifold of dimension m with a Lorentz tensor field. A restricted homotopy theory on M is defined in which the homotopy is always timelike with respect to the Lorentz tensor field. The homotopy so restricted is called timelike homotopy, and the timelike homotopy groups are denoted [tau]raised n, lowered q. The construction of the timelike homotopy groups is then applied to fibre bundles over M, the bundle space E and the fibre F both admitting Lorentz structures. The bundle of timelike homotopy groups E([tau][raised n, lowered q]) is then used as the coefficient group for a cohomology theory. Obstructions to the construction of sections of E are defined, and it is shown that the obstructions determine an element of the cohomology group H[raised n+l](M,E([tau][raised n, lowered q])). It is then shown that if E is the subbundle of the tangent bundle consisting of unit timelike vectors, then this cohomology class in dimension one vanishes if and only if M is time orientable. Finally, the non-vanishing of the m-dimensional cohomology class is shown to be a necessary and sufficient condition for the existence of a singularity in the tensor field.