Alphabeta MathDifferential Topology
Pathway
Pages are grouped by how many dependency steps into this group they sit. Everything a page needs from this group appears above it.
Level 0
6 pagesThis page develops the holonomy of a regular foliation: the way nearby leaves are compared along a leafwise path, the algebraic structures that record it, and the two global constructions that make it computable.
9 definitions, 7 lemmas, 4 propositions, 1 theorem, 1 remarkExamples & counterexamples →This page fixes the local Morse-theory vocabulary for smooth real-valued functions.
5 definitions, 6 lemmas, 2 theorems, 4 corollaries, 1 remarkExamples & counterexamples →This page counts intersections of complementary-dimensional objects.
4 definitions, 7 lemmas, 1 proposition, 4 theorems, 3 corollaries, 1 remarkExamples & counterexamples →This page develops the differentiable-topology machinery behind finite triangulations that respect extra structure: corners, boundary data, and a supplied bundle projection over a fixed base triangulation.
1 definition, 4 lemmas, 2 theoremsExamples & counterexamples →This page builds geometric cobordism from the ground up: a bordism is a compact manifold with boundary carrying a decomposition of its boundary into an incoming and an…
6 definitions, 6 lemmas, 3 propositions, 3 theorems, 1 remarkExamples & counterexamples →This page fixes the differential-topology side of the Thom construction.
4 definitions, 7 lemmas, 3 propositions, 1 theorem, 2 remarksExamples & counterexamples →
Level 1
5 pagesThis page turns the characteristic classes of the algebraic-topology pages into numerical invariants of closed manifolds and reads off what they see of cobordism.
9 lemmas, 1 proposition, 4 theorems, 1 corollary, 1 remarkExamples & counterexamples →The descending equation fixes the trajectory orientation. Under Countable Choice (AC ω), compactness supplies flow completeness; tail-limit compactness and critical-point…
6 definitions, 7 lemmas, 2 propositions, 2 theorems, 2 corollaries, 1 remarkExamples & counterexamples →This page pairs the geometric intersection count of complementary-dimensional cycles with the algebraic-topology pairing on cohomology.
2 definitions, 6 lemmas, 2 propositions, 2 theorems, 2 corollaries, 3 remarksExamples & counterexamples →This page turns the local Morse vocabulary from the previous page into global genericity statements.
5 lemmas, 1 proposition, 6 theorems, 1 corollary, 2 remarksExamples & counterexamples →- Pontryagin Thom and Framed Cobordism20 results
This page develops the Pontryagin–Thom construction and the framed cobordism relation on closed submanifolds of a closed smooth manifold X.
6 definitions, 10 lemmas, 1 proposition, 2 theorems, 1 remarkExamples & counterexamples →
Level 2
4 pagesFix the descending convention dotγ=-grad g f. A point-marked connecting trajectory is first a transverse stable--unstable intersection; quotienting by time is then justified…
4 definitions, 10 lemmas, 3 propositions, 5 theorems, 1 corollary, 2 remarksExamples & counterexamples →A compact regular band admits a controlled flow and a product description.
3 definitions, 7 lemmas, 2 propositions, 2 theorems, 6 corollaries, 1 remarkExamples & counterexamples →- The Hirzebruch Signature Theorem28 results
This page develops the Hirzebruch signature theorem: on every closed oriented smooth 4k-manifold the signature, defined as the inertia difference of the middle-dimensional…
6 definitions, 15 lemmas, 3 theorems, 3 corollaries, 1 remarkExamples & counterexamples → - The Hopf Degree Theorem20 results
Assuming AC ω and m≥1, this page classifies maps from a nonempty closed connected m-manifold to the m-sphere by degree.
3 definitions, 9 lemmas, 4 theorems, 2 corollaries, 2 remarksExamples & counterexamples →
Level 3
2 pagesAssuming Countable Choice, this page turns a compact cobordism with collared faces into a finite handle presentation and records how the presentation depends on the data.
4 definitions, 17 lemmas, 2 propositions, 5 theorems, 1 remarkExamples & counterexamples →Assuming the Axiom of Choice, on a closed manifold a Morse--Smale pair has finitely many critical points, and the mod-two Morse chain group in each degree is the free ℤ/2-module on the critical points of that index.
6 definitions, 5 lemmas, 4 theorems, 1 corollary, 2 remarksExamples & counterexamples →
Level 4
3 pagesThis page develops the homotopy-theoretic picture of immersion theory.
7 definitions, 18 lemmas, 2 theorems, 1 corollary, 2 remarksExamples & counterexamples →This page records the elementary modifications of a handle presentation: introduction and cancellation of a complementary pair of consecutive indices, handle slides, and the…
3 definitions, 10 lemmas, 2 propositions, 2 theorems, 2 remarksExamples & counterexamples →- Smooth Surgery Traces and Handle Trading17 results
This page develops a single surgery step in the smooth category, from its exact input data to its trace and its inverse.
5 definitions, 6 lemmas, 2 propositions, 2 theorems, 2 remarksExamples & counterexamples →
Level 5
3 pagesThis page turns the handle-theoretic sublevel filtration into the classical numerical Morse inequalities.
3 definitions, 8 lemmas, 2 propositions, 1 theorem, 4 corollaries, 1 remarkExamples & counterexamples →- Regular Homotopy and Sphere Eversion15 results
This page applies the Smale--Hirsch theory of formal immersions.
2 definitions, 6 lemmas, 1 proposition, 3 theorems, 3 remarksExamples & counterexamples → The Whitney trick has three separate inputs: a boundary circle that contracts, an embedded disk whose interior avoids the sheets, and an admissible normal frame that extends over that disk.
3 definitions, 18 lemmas, 1 proposition, 4 theorems, 1 corollary, 2 remarksExamples & counterexamples →
Level 6
4 pagesThis page is the ambient side of embedding theory beyond the Whitney existence theorems.
3 definitions, 11 lemmas, 1 proposition, 1 theorem, 3 corollaries, 2 remarksExamples & counterexamples →This page proves that Morse homology is an invariant of the underlying closed manifold.
6 definitions, 11 lemmas, 1 proposition, 7 theorems, 1 corollary, 1 remarkExamples & counterexamples →- The Smooth H Cobordism Theorem21 results
This page develops the smooth h-cobordism theorem for boundary dimension at least five, from the symmetric definition of an h-cobordism through the handle-theoretic normal…
2 definitions, 12 lemmas, 2 propositions, 2 theorems, 2 corollaries, 1 remarkExamples & counterexamples → The page develops the local index of an isolated zero of a smooth vector field, its chart and ball invariance, the determinant formula for nondegenerate zeros, additivity…
4 definitions, 11 lemmas, 2 propositions, 4 theorems, 5 corollaries, 1 remarkExamples & counterexamples →
Level 7
4 pagesThis page turns the characteristic classes of the published algebraic-topology pages into immersion and embedding tests for closed smooth manifolds.
2 definitions, 10 lemmas, 3 propositions, 1 theorem, 5 corollaries, 2 remarksExamples & counterexamples →This page constructs the quaternionic clutching bundles ξ h,j over S⁴ and proves that their unit sphere bundles M h,j are smooth manifolds homeomorphic to S⁷ when the Euler number h+j equals ±1.
8 definitions, 25 lemmas, 4 theorems, 2 corollaries, 2 remarksExamples & counterexamples →This page develops the fixed point index of a smooth self-map and the Lefschetz–Hopf index formula that computes it from rational homology.
4 definitions, 15 lemmas, 1 proposition, 3 theorems, 2 corollaries, 2 remarksExamples & counterexamples →This page refines the simply connected h-cobordism theorem to the non-simply-connected case by attaching the algebraic obstruction of the published Whitehead-group page to the handle calculus of the preceding pair.
2 definitions, 12 lemmas, 1 proposition, 3 theorems, 1 corollary, 2 remarksExamples & counterexamples →
Level 8
1 pageLocal Reeb stability describes a compact finite-holonomy leaf by its holonomy cover and a finite transverse action.
8 definitions, 25 lemmas, 3 propositions, 4 theorems, 2 corollaries, 2 remarksExamples & counterexamples →
Level 9
1 pageThis page brings together two strands that meet on a codimension-one foliation: the secondary characteristic classes of the foliation itself, and the local planar and…
6 definitions, 40 lemmas, 2 theorems, 1 corollary, 1 remarkExamples & counterexamples →
Level 10
1 pageThis page develops the classical Novikov theory of codimension-one foliations of closed oriented three-manifolds through the tautness and vanishing-cycle route of Calegari, Novikov and Ranz.
7 definitions, 39 lemmas, 2 propositions, 1 theorem, 1 corollary, 2 remarksExamples & counterexamples →