On a Lorentzian manifold, certain curves are distinguished as timelike. A timelike homotopy between two timelike curves is a homotopy

Homotopy

In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...

 such that each intermediate curve is timelike. No closed timelike curve

Closed timelike curve

In mathematical physics, a closed timelike curve is a worldline in a Lorentzian manifold, of a material particle in spacetime that is "closed," returning to its starting point...

 (CTC) on a Lorentzian manifold is timelike homotopic to a point (that is, null timelike homotopic); such a manifold is therefore said to be multiply connected by timelike curves (or timelike multiply connected). A manifold such as the 3-sphere

3-sphere

In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...

 can be simply connected (by any type of curve), and at the same time be timelike multiply connected. Equivalence classes of timelike homotopic curves define their own fundamental group, as noted by Smith (1967). A smooth topological feature which prevents a CTC from being deformed to a point may be called a timelike topological feature.