The geometrical techniques in piece-wise linear
topology are extended to the situation -where one has
a projection which must be preserved. These include
general position, sunny collapsing and engulfing .
They are employed to study the relative homotopy
groups of the simplicial spaces of concordances and
isotopies and, consequently, give a proof of a para-
meterized version of Hudson1s concordance implies
isotopy theorem.
Received by the editors March 25, 197^
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