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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 57R: Differential topology

Introduction

History

Applications and related fields

For application to the classification of real division algebras see the division ring FAQ.

For foundational questions of differentiable manifolds, See 58Axx; for infinite-dimensional manifolds, See 58Bxx

Specific manifolds may be treated as Lie groups, etc; for example the classical groups (viewed as geometric objects) are considered in 51N30.

Subfields

• 57R05: Triangulating
• 57R10: Smoothing
• 57R12: Smooth approximations
• 57R15: Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
• 57R17: Symplectic and contact topology [new in 2000]
• 57R19: Algebraic topology on manifolds
• 57R20: Characteristic classes and numbers
• 57R22: Topology of vector bundles and fiber bundles, See Also 55Rxx
• 57R25: Vector fields, frame fields
• 57R27: Controllability of vector fields on C^\infty and real-analytic manifolds, See also 49Qxx, 58F40, 93B05
• 57R30: Foliations; geometric theory
• 57R32: Classifying spaces for foliations; Gelfand-Fuks cohomology, See also 58H10
• 57R35: Differentiable mappings
• 57R40: Embeddings
• 57R42: Immersions
• 57R45: Singularities of differentiable mappings
• 57R50: Diffeomorphisms
• 57R52: Isotopy
• 57R55: Differentiable structures
• 57R56: Topological quantum field theories [new in 2000]
• 57R57: Applications of global analysis to structures on manifolds, See also 58-XX
• 57R58: Floer homology [new in 2000]
• 57R60: Homotopy spheres, Poincaré conjecture
• 57R65: Surgery and handlebodies
• 57R67: Surgery obstructions, Wall groups, See also 19J25
• 57R70: Critical points and critical submanifolds
• 57R75: O- and SO-cobordism
• 57R77: Complex cobordism (U- and SU-cobordism), See also 55N22
• 57R80: h- and s-cobordism
• 57R85: Equivariant cobordism
• 57R90: Other types of cobordism See 55N22
• 57R91: Equivariant algebraic topology of manifolds
• 57R95: Realizing cycles by submanifolds
• 57R99: None of the above but in this section

Parent field: 57: Manifolds and Cell complexes

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Software and tables

Other web sites with this focus

Selected topics at this site

• Thinking about the definition of a manifold
• What is the gradient (in differential topology)?
• Extending maps from submanifolds of R^k to open neighborhoods of R^k
• What is the connected sum of two manifolds?
• Is the smooth image of a manifold still a manifold? (no)
• Review of the "long line" (the imposter manifold), along with some questions.
• Can you determine analytic functions on a manifold from the values on a set with an accumulation point? (no -- read to see djr mix up results from complex analysis with real manifolds!)
• Are C^\infty manifolds real-analytic?
• Are most manifolds hyperbolic?
• Finding a manifold with boundary RP^n
• There are only finitely many subspaces of C^\infty(R) invariant under diff(R)
• What is the Poincaré sphere? The Poincaré conjecture?
• Decidability of classification questions in topology, e.g. Poincare conjecture
• Classification of 4-dimensional manifolds (up to homeo- or diffeo-morphism), and applications to mathematical physics.
• Pointers to understanding the construction of exotic differentiable structures on R^4.
• Constructing exotic differentiable structures on the 7-sphere.
• Exotic differentiable structures on spheres
• What is the Arf invariant for mod-2 quadratic spaces (e.g. the middle dimensional homology group)
• Heegaard splittings of 3-manifolds
• Reeb, Hopf foliations of S^3
• Some further comments about diffeomorphisms of spheres and balls is also available.
• A couple of questions from physicists regarding the 7-dimensional sphere and other 7-manifolds.
• How different is the real-analytic category from the C^\infty category (for real manifolds)?
• A little about Stein manifolds
• Is there a complex structure on S^6?
• What is Differential Geometry; how does it differ from differential topology? May manifolds always be embedded into Euclidean space?
• Embedding dimensions of compact smooth manifolds into Euclidean space
• Spaces which are homeomorphic but not diffeomorphic (etc)
• [Daniel Henry Gottlieb] Use vector fields to prove all the classical theorems! (Gauss-Bonnet, Jordan Curve, etc.)
• Difference between PL and differentiable manifolds.
• So which manifolds are triangularizable?
• Old (1970s) citations for triangularizability of manifolds

The Nash embedding theorem states that a Riemannian manifold embeds in some R^n isometrically. Here are some variants:

• Given a real-analytic manifold, does it have an analytic isometric embedding in some R^n? (yes)
• The Morrey-Grauert theorem: any real-analytic manifold admits a real-analytic embedding into some R^n.
• Given a Riemannian manifold with a group action on it (by isometries), is there an equivariant isometric embedding into some R^n? (yes if the manifold is compact, not necessarily if otherwise)
• What is an immersion? (Computations for a function defined on the projective plane).
• Topology of the spaces of automorphisms (invertible self-maps, in various categories) of S^3
• Odd-dimensional projective spaces are orientable.
• What is a pseudomanifold?
• Classifying almost-complex structures in vector-spaces
• What is the Prüfer manifold?
• Chain complex of differential forms
• Poincare duality and the cup-product mapping on cohomology of a manifold
• History of manifolds :-)

Spheres are great places to think about vector fields and so on; see the sphere FAQ.