Oriented smooth manifolds and oriented charts

Depends on

• Determinant-line orientations of finite-dimensional real vector spaces
• Tangent and cotangent bundles extend over a boundary
• The exterior power bundle of the cotangent bundle

Used by

• Mod-two evenness does not by itself supply a Whitney move Corollary
• Symplectic manifolds have a canonical orientation and volume form Corollary
• Same-sign intersection points cannot be cancelled orientedly Counterexample
• Attaching-belt intersection matrix of adjacent-index handles Definition
• Chart integral with its orientation sign Definition
• Coherence of Seifert circles and the height of a diagram Definition
• Hermitian metric and L² pairing on a compact Riemann surface Definition
• Local orientation sign of a regular preimage Definition
• Orientable manifolds Definition
• Orientation-preserving parametrizations Definition
• Oriented links in the three-sphere and ambient isotopy Definition
• Oriented Riemannian surface and positive quarter-turn Definition
• Oriented smooth cobordism Definition
• Pontryagin numbers of a closed oriented manifold Definition
• Positive volume form on an oriented manifold Definition
• Regular oriented surface regions with corners Definition
• Riemannian volume form on an oriented manifold Definition
• Surgery trace cobordism Definition
• The A-hat form and relative Chern character Definition
• The frame bundle of a smooth manifold Definition
• The framing sign of a zero-dimensional regular preimage Definition
• The local oriented intersection sign Definition
• The middle-handle intersection matrix of an h-cobordism Definition
• The oriented intersection number Definition
• The reduced degree of a map into the 0-sphere Definition
• A circle is the boundary of a disk Example
• A local Whitney move in Euclidean space Example
• Collapsing k oriented disks realizes degree k Example
• Oppositely signed intersections of two three-manifolds in a simply connected six-manifold Example
• Orientation reversal negates Pontryagin numbers Example
• Signed points give the oriented zero-bordism invariant Example
• The Mobius band is nonorientable although its boundary circle is orientable Example
• The pair of pants is a cobordism realizing addition of circles Example
• The product orientation on a torus Example
• Every manifold is orientable False statement
• False: density measures require an orientation False statement
• A co-oriented closed transversal detects nonvanishing rational homology of a compact leaf Lemma
• A smooth local diffeomorphism lifts canonically to the orientation double cover Lemma
• Algebraic cancellation does not yet give geometric cancellation Lemma
• Every integer is realized by a map to the sphere Lemma

…and 14 more results.

Dependency tree · two levels

13 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)