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On the brown walls of the flattened, overlapping leaves of the sturdy fibered guano there was a picture in color of the Sacred Heart of Jesus and another of the Virgin of Cobre.

The following year, Wise built on this finding to show that all Haken manifolds are virtually fibered; that is, there is a way to unroll a Haken manifold to produce a finite cover that opens up the complicated topology of the guts, resulting in a simple fibered manifold.

Thus, if a manifold is virtually Haken, then it also must be virtually fibered.

The manifold is said to be fibered because if you imagine stretching out the thickened surface so the boundary surfaces are far apart, then drawing the boundaries around to face each other before gluing them together, you can imagine that the resulting manifold is like a bracelet that has an infinitely thin surface-shaped bead at every point on the bracelet’s strand; these beads are the “fibers.”

In 2008, in what Calegari calls “an astonishing breakthrough,”Agol showed that hyperbolic three-manifolds that satisfy a certain technical condition are guaranteed to be virtually fibered.