Preposition

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.

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 ....

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 .