Product orientations

Depends on

• Determinant-line orientations of finite-dimensional real vector spaces
• Functoriality of finite-dimensional exterior powers

Used by

• The diagonal self-intersection is the Euler number of the tangent bundle Corollary
• Oriented smooth cobordism Definition
• The local oriented intersection sign Definition
• A cylinder with oppositely oriented boundary components Example
• Coordinate circles give the alternating intersection matrix of a torus Example
• Degree as an intersection with a regular value Example
• Latitude and meridian intersections on the torus Example
• Multiplicativity of the signature on products of projective spaces Example
• Signed points give the oriented zero-bordism invariant Example
• The diagonal in the two-sphere has self-intersection two Example
• The product orientation on a torus Example
• The signature of the product of two 2-spheres is zero: the hyperbolic intersection form Example
• Arbitrary metric Morse--Smale end counts form finite Morse chain complexes Lemma
• Boundary orientation of the compactified one-dimensional Morse moduli space Lemma
• Cellular boundary coefficients are the signed trajectory counts Lemma
• Cylinders give reflexivity of cobordism Lemma
• Finite flow matching gives local charts at metric-end broken trajectories Lemma
• Orientation lines orient the continuation moduli spaces compatibly with gluing Lemma
• Oriented boundary of an intersection trace has opposite end signs Lemma
• Preimage orientation agrees with the local intersection sign Lemma
• Product boundary formula for oriented manifolds Lemma
• Swapping direct summands scales oriented bases by a sign Lemma
• The fundamental class of a product is the cross product of the fundamental classes Lemma
• The fundamental-group label controls contractibility of the Whitney circle Lemma
• The local intersection sign of graph against diagonal is sign det(I-Df) Lemma
• The normal bundle of the diagonal is canonically the tangent bundle Lemma
• The signature and the L-genus agree on products of complex projective spaces Lemma
• The signed count is invariant under framed cobordism Lemma
• Unstable orientations induce orientations of the trajectory moduli spaces Lemma
• Boundary orientation of a product with at most one boundary factor Proposition
• The index of a zero is its zero-section intersection number Proposition
• Two-map intersection as a diagonal preimage Proposition
• Zero-dimensional bordism groups Proposition
• Cartesian product makes bordism a graded ring Theorem
• Disjoint union makes bordism classes abelian groups Theorem
• Intersection number under factor interchange Theorem
• The signature is multiplicative under Cartesian products Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Ioan Mărcuț, Manifolds (2017 lecture notes), §§14.5, 15.1 (standard reference, not scraped)
• Will Merry, Differential Geometry (2021), Lecture 24 (standard reference, not scraped)