Cobordisms and boundary orientations

For closed oriented n-manifolds M₀ and M₁, an oriented cobordism from M₀ to M₁ is a compact oriented smooth (n+1)-manifold W with its entire boundary identified with the disjoint union of −M₀ and M₁. The minus reverses orientation. The boundary identifications are part of the data, not just an equality of unlabeled point sets. A collar supplies a product neighborhood of the boundary, allowing two cobordisms to be glued smoothly along matching intermediate manifolds.

A cylinder gives a cobordism from a manifold to itself. Reversing a cobordism's orientation exchanges the incoming and outgoing roles, and gluing supplies transitivity of the cobordism relation. More general cobordisms can create or merge components, as a pair-of-pants surface does for circles. Thus being cobordant is weaker than being diffeomorphic. Morse functions constant on each incoming and outgoing boundary component and equal to a collar coordinate near each end have no critical points there. Their interior critical points break a compact cobordism into elementary handle transitions, connecting this global relation with the previous lesson's local critical events.

At a boundary point of an oriented manifold, a tangent basis of the boundary is positive when placing an outward transverse vector before it produces a positive basis of the full tangent space. A metric can supply an outward unit normal, but the orientation rule depends only on its outward component. For the cylinder oriented by dt followed by an oriented basis of M, the outward normal at t=1 is positive ∂t, so the boundary orientation agrees with M. At t=0 it is negative ∂t, so the induced orientation is reversed.

In a positively oriented planar region, this rule makes an outer boundary counterclockwise and an inner boundary clockwise. The interior must remain on the left when walking in the positive direction. Reusing the same counterclockwise parameterization for every boundary circle would forget the inner sign. Similarly, switching the order of factors in a product orientation can introduce a dimension-dependent sign; fixing dt first avoids that ambiguity in the cylinder formula used here.

For a compact oriented smooth (n+1)-manifold with boundary and a smooth n-form α, Stokes' theorem equates the integral of dα over W with the integral of α over its induced oriented boundary. Under a cobordism identification this becomes the outgoing integral minus the incoming integral. If α is closed on W, the two boundary integrals agree. This conclusion requires one extending form on the whole cobordism; arbitrary unrelated forms on the two ends need not have such an extension or equal integrals.

Boundary geometry also controls gluing: along the common interface, the two induced orientations must be opposite so that their boundary integrals cancel. Collars arrange the smooth structure near that interface, while the orientation convention arranges the signs. Statements about manifolds with corners need a corresponding version or smoothing step, so the definitions here use smooth boundaries. Cobordism packages these local compatibility checks into a global comparison, rather than claiming that every interpolation preserves each geometric quantity.