Novikov's conclusions do not extend to higher dimensions or noncompact manifolds

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$. Novikov's theorem is deliberately three- dimensional and codimension one. In higher dimensions the analogue fails even for codimension-one strongly symplectic foliations: Venugopalan constructs a closed $5$-manifold with a codimension-one foliation whose leaves have non-injective inclusion in the fundamental group of the ambient manifold and which admits a closed transversal that is null-homotopic in the ambient manifold. The two fundamental-group conclusions therefore fail in dimension five even in the strongly symplectic class. No higher-dimensional notion of Reeb component is defined or asserted here. The compactness hypothesis also cannot be dropped: the crossing example of the companion page removes a point from a closed three-manifold foliated by dense cylinders and produces a noncompact $3$-manifold with a Reebless foliation and a leaf whose inclusion is not ${\pi }_1$-injective, so the conclusion of Reebless leaves are pi-one-injective and transverse loops are essential fails without compactness (the leaf escapes through the puncture). Both restrictions are part of the statement and are not artefacts of the proof.

Recorded as a scope caveat with its sources. In higher dimensions the analogue fails even for codimension-one strongly symplectic foliations: Venugopalan constructs a closed $5$-manifold with a codimension-one foliation whose leaves are not ${\pi }_1$-injective and which admits a null-homotopic closed transversal. These are the two conclusions of Venugopalan’s Theorem 1; a higher-dimensional “Reeb-type component” is not part of that theorem or a definition supplied here. Compactness also cannot be dropped: the crossing example on the companion page removes a point from a closed three-manifold foliated by dense cylinders and produces a noncompact Reebless foliation with a leaf whose inclusion is not ${\pi }_1$-injective, so the conclusion of Reebless leaves are pi-one-injective and transverse loops are essential fails without compactness. Both restrictions belong to the statements and are not artefacts of the proof.