= Relative boundary rigidity from a hierarchy
A map between <compact> <connected> <orientable> irreducible <3-manifolds> with nonempty boundaries, <injective> on the <boundary> and on the <fundamental group>, is <homotopic> to a <homeomorphism> relative to the <boundary>. The <boundary> determines degree $\pm1$, and the cover corresponding to the image of the <fundamental group> consequently has one sheet. Straighten the preimage of an incompressible hierarchy <surface>, cut both manifolds, and repeat. At the terminal balls, extend the prescribed <boundary> <homeomorphisms> by coning. The <homotopies> and <homeomorphisms> glue across the straightened <surface> collars.
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