55Qxx Homotopy groups and related invariants
This subtopic introduces the core ideas in homotopy groups and related invariants, 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
55Q05 Homotopy groups, general; sets of homotopy classes
Overview
Homotopy groups, general; sets of homotopy classes. This topic focuses on homotopy groups and related invariants, including computational techniques and structure theorems.
Related Wikipedia Page
Wikipedia: Homotopy group
Useful Links
Key Ideas
- higher homotopy groups
- exact sequences and obstruction tools
- computational and stable methods
Typical Uses
Used to extract refined invariants of spaces beyond homology and to detect deep geometric features.
Applications
- Manifold topology
- Stable homotopy computations
- Homotopical obstruction theory
References
Recommended Textbooks
55Q07 Shape groups
Overview
Shape groups. This topic focuses on homotopy groups and related invariants, including computational techniques and structure theorems.
Related Wikipedia Page
Wikipedia: Homotopy group
Useful Links
Key Ideas
- higher homotopy groups
- exact sequences and obstruction tools
- computational and stable methods
Typical Uses
Used to extract refined invariants of spaces beyond homology and to detect deep geometric features.
Applications
- Manifold topology
- Stable homotopy computations
- Homotopical obstruction theory
References
Recommended Textbooks
55Q10 Stable homotopy groups
Overview
Stable homotopy groups. This topic focuses on homotopy groups and related invariants, including computational techniques and structure theorems.
Related Wikipedia Page
Wikipedia: Homotopy group
Useful Links
Key Ideas
- higher homotopy groups
- exact sequences and obstruction tools
- computational and stable methods
Typical Uses
Used to extract refined invariants of spaces beyond homology and to detect deep geometric features.
Applications
- Manifold topology
- Stable homotopy computations
- Homotopical obstruction theory
References
Recommended Textbooks
55Q15 Whitehead products and generalizations
Overview
Whitehead products and generalizations. This topic focuses on homotopy groups and related invariants, including computational techniques and structure theorems.
Related Wikipedia Page
Wikipedia: Homotopy group
Useful Links
Key Ideas
- higher homotopy groups
- exact sequences and obstruction tools
- computational and stable methods
Typical Uses
Used to extract refined invariants of spaces beyond homology and to detect deep geometric features.
Applications
- Manifold topology
- Stable homotopy computations
- Homotopical obstruction theory
References
Recommended Textbooks
55Q20 Homotopy groups of wedges, joins, and simple spaces
Overview
Homotopy groups of wedges, joins, and simple spaces. This topic focuses on homotopy groups and related invariants, including computational techniques and structure theorems.
Related Wikipedia Page
Wikipedia: Homotopy group
Useful Links
Key Ideas
- higher homotopy groups
- exact sequences and obstruction tools
- computational and stable methods
Typical Uses
Used to extract refined invariants of spaces beyond homology and to detect deep geometric features.
Applications
- Manifold topology
- Stable homotopy computations
- Homotopical obstruction theory
References
Recommended Textbooks
55Q25 Hopf invariants
Overview
Hopf invariants. This topic focuses on homotopy groups and related invariants, including computational techniques and structure theorems.
Related Wikipedia Page
Wikipedia: Homotopy group
Useful Links
Key Ideas
- higher homotopy groups
- exact sequences and obstruction tools
- computational and stable methods
Typical Uses
Used to extract refined invariants of spaces beyond homology and to detect deep geometric features.
Applications
- Manifold topology
- Stable homotopy computations
- Homotopical obstruction theory
References
Recommended Textbooks
55Q35 Operations in homotopy groups
Overview
Operations in homotopy groups. This topic focuses on homotopy groups and related invariants, including computational techniques and structure theorems.
Related Wikipedia Page
Wikipedia: Homotopy group
Useful Links
Key Ideas
- higher homotopy groups
- exact sequences and obstruction tools
- computational and stable methods
Typical Uses
Used to extract refined invariants of spaces beyond homology and to detect deep geometric features.
Applications
- Manifold topology
- Stable homotopy computations
- Homotopical obstruction theory
References
Recommended Textbooks
55Q40 Homotopy groups of spheres
Overview
Homotopy groups of spheres. This topic focuses on homotopy groups and related invariants, including computational techniques and structure theorems.
Related Wikipedia Page
Wikipedia: Homotopy group
Useful Links
Key Ideas
- higher homotopy groups
- exact sequences and obstruction tools
- computational and stable methods
Typical Uses
Used to extract refined invariants of spaces beyond homology and to detect deep geometric features.
Applications
- Manifold topology
- Stable homotopy computations
- Homotopical obstruction theory
References
Recommended Textbooks
55Q45 Stable homotopy of spheres
Overview
Stable homotopy of spheres. This topic focuses on homotopy groups and related invariants, including computational techniques and structure theorems.
Related Wikipedia Page
Wikipedia: Homotopy group
Useful Links
Key Ideas
- higher homotopy groups
- exact sequences and obstruction tools
- computational and stable methods
Typical Uses
Used to extract refined invariants of spaces beyond homology and to detect deep geometric features.
Applications
- Manifold topology
- Stable homotopy computations
- Homotopical obstruction theory
References
Recommended Textbooks
55Q50 $J$-morphism
Overview
$J$-morphism. This topic focuses on homotopy groups and related invariants, including computational techniques and structure theorems.
Related Wikipedia Page
Wikipedia: Homotopy group
Useful Links
Key Ideas
- higher homotopy groups
- exact sequences and obstruction tools
- computational and stable methods
Typical Uses
Used to extract refined invariants of spaces beyond homology and to detect deep geometric features.
Applications
- Manifold topology
- Stable homotopy computations
- Homotopical obstruction theory
References
Recommended Textbooks
55Q51 Stable homotopy of complex projective space
Overview
Stable homotopy of complex projective space. This topic focuses on homotopy groups and related invariants, including computational techniques and structure theorems.
Related Wikipedia Page
Wikipedia: Homotopy group
Useful Links
Key Ideas
- higher homotopy groups
- exact sequences and obstruction tools
- computational and stable methods
Typical Uses
Used to extract refined invariants of spaces beyond homology and to detect deep geometric features.
Applications
- Manifold topology
- Stable homotopy computations
- Homotopical obstruction theory
References
Recommended Textbooks
55Q52 Homotopy groups of Thom spectra, cobordism, and Morava $K$-theory
Overview
Homotopy groups of Thom spectra, cobordism, and Morava $K$-theory. This topic focuses on homotopy groups and related invariants, including computational techniques and structure theorems.
Related Wikipedia Page
Wikipedia: Homotopy group
Useful Links
Key Ideas
- higher homotopy groups
- exact sequences and obstruction tools
- computational and stable methods
Typical Uses
Used to extract refined invariants of spaces beyond homology and to detect deep geometric features.
Applications
- Manifold topology
- Stable homotopy computations
- Homotopical obstruction theory
References
Recommended Textbooks
55Q55 Cohomotopy groups
Overview
Cohomotopy groups. This topic focuses on homotopy groups and related invariants, including computational techniques and structure theorems.
Related Wikipedia Page
Wikipedia: Homotopy group
Useful Links
Key Ideas
- higher homotopy groups
- exact sequences and obstruction tools
- computational and stable methods
Typical Uses
Used to extract refined invariants of spaces beyond homology and to detect deep geometric features.
Applications
- Manifold topology
- Stable homotopy computations
- Homotopical obstruction theory
References
Recommended Textbooks
55Q70 Homotopy groups of special types
Overview
Homotopy groups of special types. This topic focuses on homotopy groups and related invariants, including computational techniques and structure theorems.
Related Wikipedia Page
Wikipedia: Homotopy group
Useful Links
Key Ideas
- higher homotopy groups
- exact sequences and obstruction tools
- computational and stable methods
Typical Uses
Used to extract refined invariants of spaces beyond homology and to detect deep geometric features.
Applications
- Manifold topology
- Stable homotopy computations
- Homotopical obstruction theory
References
Recommended Textbooks
55Q91 Equivariant homotopy groups
Overview
Equivariant homotopy groups. This topic focuses on homotopy groups and related invariants, including computational techniques and structure theorems.
Related Wikipedia Page
Wikipedia: Homotopy group
Useful Links
Key Ideas
- higher homotopy groups
- exact sequences and obstruction tools
- computational and stable methods
Typical Uses
Used to extract refined invariants of spaces beyond homology and to detect deep geometric features.
Applications
- Manifold topology
- Stable homotopy computations
- Homotopical obstruction theory
References
Recommended Textbooks