Induced boundary orientation

Depends on

• Determinant-line orientations of finite-dimensional real vector spaces
• Inward, outward, and boundary-tangent vectors
• The boundary tangent space is the boundary-tangent hyperplane

Used by

• A cycle has zero algebraic intersection with a bounding cycle Corollary
• Sphere self-maps are homotopic exactly when their degrees agree Corollary
• Sphere self-maps of degree ±1 are exactly the homotopy equivalences Corollary
• A reflected sphere embedding is regularly homotopic but not isotopic to the standard one Counterexample
• Vanishing stable characteristic classes do not make two embeddings isotopic Counterexample
• Attaching-belt intersection matrix of adjacent-index handles Definition
• Coherence of Seifert circles and the height of a diagram Definition
• Oriented smooth cobordism Definition
• Regular oriented surface regions with corners Definition
• Standard orientation of the affine simplex Definition
• Surgery trace cobordism Definition
• The reduced degree of a map into the 0-sphere Definition
• A circle is the boundary of a disk Example
• A cylinder with oppositely oriented boundary components Example
• A sphere reflection has degree minus one Example
• Boundary orientation of the unit sphere by the outward normal Example
• Collapsing k oriented disks realizes degree k Example
• Degree of a reflection of a sphere Example
• Positive-dimensional real projective space is orientable exactly in odd dimension Example
• Signed points give the oriented zero-bordism invariant Example
• The boundary of an oriented interval Example
• The pair of pants is a cobordism realizing addition of circles Example
• Two unoriented points bound an interval Example
• Boundary orientation is inward-normal-first False statement
• An ambient isotopy preserves the orientation of an invariant round sphere Lemma
• Arbitrary metric Morse--Smale end counts form finite Morse chain complexes Lemma
• Bordant cycles have equal intersection numbers Lemma
• Boundary orientation of the compactified one-dimensional Morse moduli space Lemma
• Cellular boundary coefficients are the signed trajectory counts Lemma
• Collar gluing and seam smoothing give transitivity Lemma
• Compact-support Stokes on the upper half-space Lemma
• Cylinders give reflexivity of cobordism Lemma
• Every integer is realized by a map to the sphere Lemma
• Finite tangent index count and inward boundary sum Lemma
• Handle boundary coefficients are attaching-belt intersection numbers Lemma
• Orientation lines orient the continuation moduli spaces compatibly with gluing Lemma
• Oriented boundary counts of a compact oriented 1-manifold cancel Lemma
• Oriented boundary of an intersection trace has opposite end signs Lemma
• Product boundary formula for oriented manifolds Lemma
• Reversing a cobordism gives symmetry Lemma

…and 14 more results.

Dependency tree · two levels

8 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)