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The definition starts out with:
A 1-mesh is a framed contractible -manifold…
Since the condition that it be stratified appears only further down the sentence, it is not clear that “framed manifold” here is not (as far as I understand you) meant in the usual sense.
(See also another comment here.)
It is actually meant in the usual sense here! But good point, I’ll try to clarify.
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