55Rxx Fiber spaces and bundles
This subtopic introduces the core ideas in fiber spaces and bundles, 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
55R05 Fiber spaces in algebraic topology
Overview
Fiber spaces in algebraic topology. This topic studies fiber spaces and bundles, including classification theory, characteristic classes, and bundle constructions.
Related Wikipedia Page
Wikipedia: Fiber bundle
Useful Links
Key Ideas
- bundle classification
- transition functions and structure groups
- associated and principal bundle frameworks
Typical Uses
Used to model geometric and topological structures with local triviality and global twisting.
Applications
- Gauge theory
- Differential geometry
- Characteristic class computations
References
Recommended Textbooks
55R10 Fiber bundles in algebraic topology
Overview
Fiber bundles in algebraic topology. This topic studies fiber spaces and bundles, including classification theory, characteristic classes, and bundle constructions.
Related Wikipedia Page
Wikipedia: Fiber bundle
Useful Links
Key Ideas
- bundle classification
- transition functions and structure groups
- associated and principal bundle frameworks
Typical Uses
Used to model geometric and topological structures with local triviality and global twisting.
Applications
- Gauge theory
- Differential geometry
- Characteristic class computations
References
Recommended Textbooks
55R12 Transfer for fiber spaces and bundles
Overview
Transfer for fiber spaces and bundles. This topic studies fiber spaces and bundles, including classification theory, characteristic classes, and bundle constructions.
Related Wikipedia Page
Wikipedia: Fiber bundle
Useful Links
Key Ideas
- bundle classification
- transition functions and structure groups
- associated and principal bundle frameworks
Typical Uses
Used to model geometric and topological structures with local triviality and global twisting.
Applications
- Gauge theory
- Differential geometry
- Characteristic class computations
References
Recommended Textbooks
55R15 Classification of fiber spaces or bundles in algebraic topology
Overview
Classification of fiber spaces or bundles in algebraic topology. This topic studies fiber spaces and bundles, including classification theory, characteristic classes, and bundle constructions.
Related Wikipedia Page
Wikipedia: Fiber bundle
Useful Links
Key Ideas
- bundle classification
- transition functions and structure groups
- associated and principal bundle frameworks
Typical Uses
Used to model geometric and topological structures with local triviality and global twisting.
Applications
- Gauge theory
- Differential geometry
- Characteristic class computations
References
Recommended Textbooks
55R20 Spectral sequences and homology of fiber spaces in algebraic topology
Overview
Spectral sequences and homology of fiber spaces in algebraic topology. This topic studies fiber spaces and bundles, including classification theory, characteristic classes, and bundle constructions.
Related Wikipedia Page
Wikipedia: Fiber bundle
Useful Links
Key Ideas
- bundle classification
- transition functions and structure groups
- associated and principal bundle frameworks
Typical Uses
Used to model geometric and topological structures with local triviality and global twisting.
Applications
- Gauge theory
- Differential geometry
- Characteristic class computations
References
Recommended Textbooks
55R25 Sphere bundles and vector bundles in algebraic topology
Overview
Sphere bundles and vector bundles in algebraic topology. This topic studies fiber spaces and bundles, including classification theory, characteristic classes, and bundle constructions.
Related Wikipedia Page
Wikipedia: Fiber bundle
Useful Links
Key Ideas
- bundle classification
- transition functions and structure groups
- associated and principal bundle frameworks
Typical Uses
Used to model geometric and topological structures with local triviality and global twisting.
Applications
- Gauge theory
- Differential geometry
- Characteristic class computations
References
Recommended Textbooks
55R35 Classifying spaces of groups and $H$-spaces in algebraic topology
Overview
Classifying spaces of groups and $H$-spaces in algebraic topology. This topic studies fiber spaces and bundles, including classification theory, characteristic classes, and bundle constructions.
Related Wikipedia Page
Wikipedia: Fiber bundle
Useful Links
Key Ideas
- bundle classification
- transition functions and structure groups
- associated and principal bundle frameworks
Typical Uses
Used to model geometric and topological structures with local triviality and global twisting.
Applications
- Gauge theory
- Differential geometry
- Characteristic class computations
References
Recommended Textbooks
55R37 Maps between classifying spaces in algebraic topology
Overview
Maps between classifying spaces in algebraic topology. This topic studies fiber spaces and bundles, including classification theory, characteristic classes, and bundle constructions.
Related Wikipedia Page
Wikipedia: Fiber bundle
Useful Links
Key Ideas
- bundle classification
- transition functions and structure groups
- associated and principal bundle frameworks
Typical Uses
Used to model geometric and topological structures with local triviality and global twisting.
Applications
- Gauge theory
- Differential geometry
- Characteristic class computations
References
Recommended Textbooks
55R40 Homology of classifying spaces and cohomology of groups in algebraic topology
Overview
Homology of classifying spaces and cohomology of groups in algebraic topology. This topic studies fiber spaces and bundles, including classification theory, characteristic classes, and bundle constructions.
Related Wikipedia Page
Wikipedia: Fiber bundle
Useful Links
Key Ideas
- bundle classification
- transition functions and structure groups
- associated and principal bundle frameworks
Typical Uses
Used to model geometric and topological structures with local triviality and global twisting.
Applications
- Gauge theory
- Differential geometry
- Characteristic class computations
References
Recommended Textbooks
55R45 Homology and homotopy of $BO$ and $BU$; Bott periodicity
Overview
Homology and homotopy of $BO$ and $BU$; Bott periodicity. This topic studies fiber spaces and bundles, including classification theory, characteristic classes, and bundle constructions.
Related Wikipedia Page
Wikipedia: Fiber bundle
Useful Links
Key Ideas
- bundle classification
- transition functions and structure groups
- associated and principal bundle frameworks
Typical Uses
Used to model geometric and topological structures with local triviality and global twisting.
Applications
- Gauge theory
- Differential geometry
- Characteristic class computations
References
Recommended Textbooks
55R50 Stable classes of vector space bundles, $K$-theory and generalizations
Overview
Stable classes of vector space bundles, $K$-theory and generalizations. This topic studies fiber spaces and bundles, including classification theory, characteristic classes, and bundle constructions.
Related Wikipedia Page
Wikipedia: Fiber bundle
Useful Links
Key Ideas
- bundle classification
- transition functions and structure groups
- associated and principal bundle frameworks
Typical Uses
Used to model geometric and topological structures with local triviality and global twisting.
Applications
- Gauge theory
- Differential geometry
- Characteristic class computations
References
Recommended Textbooks
55R55 Fiberings with singularities
Overview
Fiberings with singularities. This topic studies fiber spaces and bundles, including classification theory, characteristic classes, and bundle constructions.
Related Wikipedia Page
Wikipedia: Fiber bundle
Useful Links
Key Ideas
- bundle classification
- transition functions and structure groups
- associated and principal bundle frameworks
Typical Uses
Used to model geometric and topological structures with local triviality and global twisting.
Applications
- Gauge theory
- Differential geometry
- Characteristic class computations
References
Recommended Textbooks
55R60 Microbundles and block bundles
Overview
Microbundles and block bundles. This topic studies fiber spaces and bundles, including classification theory, characteristic classes, and bundle constructions.
Related Wikipedia Page
Wikipedia: Fiber bundle
Useful Links
Key Ideas
- bundle classification
- transition functions and structure groups
- associated and principal bundle frameworks
Typical Uses
Used to model geometric and topological structures with local triviality and global twisting.
Applications
- Gauge theory
- Differential geometry
- Characteristic class computations
References
Recommended Textbooks
55R65 Generalizations of fiber spaces and bundles in algebraic topology
Overview
Generalizations of fiber spaces and bundles in algebraic topology. This topic studies fiber spaces and bundles, including classification theory, characteristic classes, and bundle constructions.
Related Wikipedia Page
Wikipedia: Fiber bundle
Useful Links
Key Ideas
- bundle classification
- transition functions and structure groups
- associated and principal bundle frameworks
Typical Uses
Used to model geometric and topological structures with local triviality and global twisting.
Applications
- Gauge theory
- Differential geometry
- Characteristic class computations
References
Recommended Textbooks
55R70 Fibrewise topology
Overview
Fibrewise topology. This topic studies fiber spaces and bundles, including classification theory, characteristic classes, and bundle constructions.
Related Wikipedia Page
Wikipedia: Fiber bundle
Useful Links
Key Ideas
- bundle classification
- transition functions and structure groups
- associated and principal bundle frameworks
Typical Uses
Used to model geometric and topological structures with local triviality and global twisting.
Applications
- Gauge theory
- Differential geometry
- Characteristic class computations
References
Recommended Textbooks
55R80 Discriminants in algebraic topology
Overview
Discriminants in algebraic topology. This topic studies fiber spaces and bundles, including classification theory, characteristic classes, and bundle constructions.
Related Wikipedia Page
Wikipedia: Fiber bundle
Useful Links
Key Ideas
- bundle classification
- transition functions and structure groups
- associated and principal bundle frameworks
Typical Uses
Used to model geometric and topological structures with local triviality and global twisting.
Applications
- Gauge theory
- Differential geometry
- Characteristic class computations
References
Recommended Textbooks
55R91 Equivariant fiber spaces and bundles
Overview
Equivariant fiber spaces and bundles. This topic studies fiber spaces and bundles, including classification theory, characteristic classes, and bundle constructions.
Related Wikipedia Page
Wikipedia: Fiber bundle
Useful Links
Key Ideas
- bundle classification
- transition functions and structure groups
- associated and principal bundle frameworks
Typical Uses
Used to model geometric and topological structures with local triviality and global twisting.
Applications
- Gauge theory
- Differential geometry
- Characteristic class computations
References
Recommended Textbooks