57Rxx Differential topology
This subtopic introduces the core ideas in differential topology, including foundational concepts, standard methods, and the main questions used to organize the area. Typical uses include building mathematical background, framing related research problems, and supporting applications in neighboring fields where these concepts provide useful structure.
Specific topics
57R05 Triangulating manifolds
Overview
Triangulating manifolds. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R10 Smoothing in differential topology
Overview
Smoothing in differential topology. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R12 Smooth approximations
Overview
Smooth approximations. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R15 Specialized structures on manifolds
Overview
Specialized structures on manifolds. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R17 Symplectic and contact topology
Overview
Symplectic and contact topology. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R19 Algebraic topology on manifolds and differential topology
Overview
Algebraic topology on manifolds and differential topology. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R20 Characteristic classes and numbers in differential topology
Overview
Characteristic classes and numbers in differential topology. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R22 Topology of vector bundles and fiber bundles
Overview
Topology of vector bundles and fiber bundles. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R25 Vector fields, frame fields in differential topology
Overview
Vector fields, frame fields in differential topology. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R27 Controllability of vector fields on $C^\infty$ and real-analytic manifolds
Overview
Controllability of vector fields on $C^\infty$ and real-analytic manifolds. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R30 Foliations in differential topology
Overview
Foliations in differential topology. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R32 Classifying spaces for foliations; Gelfand-Fuks cohomology
Overview
Classifying spaces for foliations; Gelfand-Fuks cohomology. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R35 Differentiable mappings in differential topology
Overview
Differentiable mappings in differential topology. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R37 Maps transversal to foliations
Overview
Maps transversal to foliations. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R38 Exotic and unusual differentiable structures on manifolds
Overview
Exotic and unusual differentiable structures on manifolds. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R40 Embeddings in differential topology
Overview
Embeddings in differential topology. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R42 Immersions in differential topology
Overview
Immersions in differential topology. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R45 Singularities of differentiable maps in differential topology
Overview
Singularities of differentiable maps in differential topology. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R50 Differential topological aspects of diffeomorphisms
Overview
Differential topological aspects of diffeomorphisms. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R52 Isotopy in differential topology
Overview
Isotopy in differential topology. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R55 Differentiable structures in differential topology
Overview
Differentiable structures in differential topology. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R56 Topological quantum field theories in differential topology
Overview
Topological quantum field theories in differential topology. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R57 Applications of global analysis to structures on manifolds
Overview
Applications of global analysis to structures on manifolds. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R58 Floer homology
Overview
Floer homology. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R60 Homotopy spheres, Poincaré conjecture
Overview
Homotopy spheres, Poincaré conjecture. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R65 Surgery and handle decompositions
Overview
Surgery and handle decompositions. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R67 Surgery obstructions, Wall groups
Overview
Surgery obstructions, Wall groups. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R70 Critical points and critical submanifolds in differential topology
Overview
Critical points and critical submanifolds in differential topology. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R75 $O$- and $SO$-cobordism
Overview
$O$- and $SO$-cobordism. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R77 Complex cobordism ($U$- and $SU$-cobordism)
Overview
Complex cobordism ($U$- and $SU$-cobordism). This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R80 $h$- and $s$-cobordism
Overview
$h$- and $s$-cobordism. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R85 Equivariant cobordism
Overview
Equivariant cobordism. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R90 Other types of cobordism
Overview
Other types of cobordism. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R91 Equivariant algebraic topology of manifolds
Overview
Equivariant algebraic topology of manifolds. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks
57R95 Realizing cycles by submanifolds
Overview
Realizing cycles by submanifolds. This topic covers differential topology, emphasizing smooth manifolds, transversality, cobordism, and smooth map singularity theory.
Related Wikipedia Page
Wikipedia: Differential topology
Useful Links
Key Ideas
- smooth structures and transversality
- cobordism and characteristic classes
- singularity and immersion/embedding theory
Typical Uses
Used to analyze global behavior of smooth maps and classify manifolds up to smooth equivalence notions.
Applications
- Geometric topology
- Morse theory
- Mathematical physics interfaces
References
Recommended Textbooks