Definition of "rel"

Since M 0 {\displaystyle M_{0}} and M 1 {\displaystyle M_{1}} are cobordant rel boundary and f 0 , f 1 {\displaystyle f_{0},f_{1}} are n-connected, the number δ := 1 2 ( r k H n M 0 − r k H n M 1 ) {\displaystyle \delta :={\frac {1}{2}}(\mathrm {rk} H_{n}M_{0}-\mathrm {rk} H_{n}M_{1})} is a nonnnegative integer.

1994, N[ikita] Netsvetaev, “Diffeomorphism Criteria for Simply Connected Even-Dimensional Manifolds”, in Oleg Viro, editor, Topology of Manifolds and Varieties (Advances in Soviet Mathematics [series] 18), American Mathematical Society, page 237

The following notations are in common use: H : f ≃ g {\displaystyle H:f\simeq g} (rel A) means that H is a homotopy rel A from f to g....

1997, Paul Selick, Introduction to Homotopy Theory (Fields Institute Monographs), American Mathematical Society, page 15

Then in n ≥ n ′ {\displaystyle n\geq n'} , there is a map h : Z → Z ′ {\displaystyle h:Z\to Z'} such that h | A = g {\displaystyle h|A=g} and g f ≃ f ′ h {\displaystyle gf\simeq f'h} rel A, so the diagram above is commutative up to homotopy rel A.

2002, Allen Hatcher, Algebraic Topology‎, Cambridge, page 355