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Let M be a simply-connected closed Poincare Duality complex of dimension n. Then M is obtained by attaching a cell of highest dimension to its (n-1)-skeleton M'. Conditions are given for when the skeletal inclusion i:M' --> M has the property that the based loops on i has a right homotopy inverse. This is an integral version of the rational statement that such a right homotopy inverse always exists provided the rational cohomology of M is not generated by a single element. New methods are developed in order to do the integral case. These lead to p-local versions and recover the full rational statement. Families for which the integral statement holds include moment-angle manifolds and quasi-toric manifolds.
Stephen Theriault (Wed,) studied this question.