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[Texts]## 57R: Differential topology |

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.

- 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.

- Thinking about the definition of a manifold
- What is the gradient (in differential topology)?
- 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)
- 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.
- What is the Arf invariant for mod-2 quadratic spaces (e.g. the middle dimensional homology group)
- Constructing exotic differentiable structures on the 7-sphere.
- Some further comments about diffeomorphisms of spheres and balls is also available.
- I've also included 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?
- 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).
- Odd-dimensional projective spaces are orientable.

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

Last modified 1999/05/12 by Dave Rusin. Mail: feedback@math-atlas.org