Preposition

1. Relative to.
   • 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 [publisher], ISBN 0821841246, page 237,

     Since $M\_0$ and $M\_1$ are cobordant rel boundary and $f\_0,\; f\_1$ are n-connected, the number $\backslash delta:=\backslash frac12(\backslash mathrm\{rk\}\; H\_nM\_0-\backslash mathrm\{rk\}H\_nM\_1)$ is a nonnnegative integer.

   • 1997, Paul Selick, Introduction to Homotopy Theory, Fields Institute Monographs [series], American Mathematical Society [publisher], ISBN 0821806904, page 15,

     The following notations are in common use: $H:f\backslash simeq\; g$ (rel A) means that H is a homotopy rel A from f to g....

   • 2002, Allen Hatcher, Algebraic Topology, Cambridge, ISBN 0521795400, page 355,

     Then in $n\backslash ge\; n\text{'}$, there is a map $h:Z\backslash to\; Z\text{'}$ such that $h|A=g$ and $gf\backslash simeq\; f\text{'}h$ rel A, so the diagram above is commutative up to homotopy rel A.