Ordinary obstruction for obstructed semi-manifolds

Let a semi-manifold be a space with possibly noninvertible transition functions, and call it obstructed when some of the usual cocycle conditions fail. Its ordinary obstruction refers to obstruction theory for extending or constructing the relevant geometric structure.

Obstructed semi-manifold obstruction conjecture. Obstructed semi-manifolds may have nonvanishing ordinary obstruction, and this obstruction can be calculated by extending standard obstruction-theoretic methods to the noninvertible case.

The source suggests extending methods associated with ordinary manifolds and supermanifolds, but does not state a resolution or provide a general construction. The status is therefore open.