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Oriented smooth cobordism

Definition

Let $M_{0}$ and $M_{1}$ be closed oriented smooth $n$ -manifolds, with orientations $o_{0}$ and $o_{1}$ . An oriented bordism from $M_{0}$ to $M_{1}$ is a bordism $(W, θ_{0}, θ_{1})$ from $M_{0}$ to $M_{1}$ in the sense of Unoriented smooth cobordism of closed manifolds, together with an orientation of $W$ , such that the induced boundary orientation (Induced boundary orientation) of the incoming face $(\partial W)_{0}$ is the negative of the supplied orientation $o_{0}$ of $M_{0}$ , and the induced boundary orientation of the outgoing face $(\partial W)_{1}$ is the supplied orientation $o_{1}$ of $M_{1}$ . Oriented manifolds are oriented cobordant when such data exist.

Equivalent collar formulation. The condition is equivalent to the requirement that, in the collar parametrisations, the embeddings $θ_{0} : [0, 1) \times M_{0} \to W$ and $θ_{1} : (- 1, 0] \times M_{1} \to W$ are orientation-preserving (Orientation-preserving parametrizations) for the product orientations of $[0, 1) \times M_{0}$ and $(- 1, 0] \times M_{1}$ (Product orientations), the intervals carrying their standard orientations. Indeed, at a point of the incoming collar the derivative of $θ_{0}$ in the interval direction is an inward normal vector, so $θ_{0}$ is orientation-preserving exactly when an outward vector followed by the image orientation of $M_{0}$ is a negative determinant of $T W$ , that is, exactly when the induced boundary orientation of $(\partial W)_{0}$ is $- o_{0}$ ; at the outgoing collar the derivative in the interval direction is an outward normal vector, so $θ_{1}$ is orientation-preserving exactly when the induced orientation of $(\partial W)_{1}$ is $o_{1}$ . The induced orientation is independent of the choice of outward vector field (Boundary orientation is independent of the outward vector field), so the condition is well posed. This is the same outward-normal-first convention that the relative fundamental class uses to define the induced boundary orientation (Relative fundamental class and boundary orientation).

Orientation reversal and empty manifolds. An orientation of a manifold is a smooth choice of a ray in each determinant line (Oriented smooth manifolds and oriented charts); the opposite orientation reverses every ray pointwise and the manifold with that orientation is written $- M$ . For a disconnected oriented manifold the reversal is taken on every component. The empty manifold carries its unique orientation and is its own negative. The definition uses no choice principle: the orientation of $W$ and the collar data are supplied, and the boundary orientation is determined by the outward-normal-first convention.

Depends on

Used by

Null-cobordant closed manifolds Definition
Unoriented and oriented bordism groups Definition
A circle is the boundary of a disk Example
Signed points give the oriented zero-bordism invariant Example
The pair of pants is a cobordism realizing addition of circles Example
Collar gluing and seam smoothing give transitivity Lemma
Cylinders give reflexivity of cobordism Lemma
Reversing a cobordism gives symmetry Lemma
Zero-dimensional bordism groups Proposition
Cartesian product makes bordism a graded ring Theorem
Disjoint union makes bordism classes abelian groups Theorem
Smooth cobordism is an equivalence relation Theorem

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Sources

Daniel S. Freed, Bordism: Old and New (lecture notes, UT Austin, Fall 2012) (standard reference, not scraped)
John Milnor and James Stasheff, Characteristic Classes (original pagination) (standard reference, not scraped)