Mathematics Branches, Topics, and Sub-Topics

A structured visual guide to the major mathematical areas and their relationships.

Search by code, branch, topic, subtopic, or a keyword from the descriptions.

55Mxx Classical topics

This subtopic introduces the core ideas in classical topics, 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

55M05 Duality in algebraic topology

Overview

Duality in algebraic topology. This topic contains classical constructions and results in algebraic topology, including foundational invariants and decomposition methods.

Related Wikipedia Page

Wikipedia: Algebraic topology

Useful Links

Key Ideas

  • classical homotopy and homology constructions
  • cellular and simplicial techniques
  • topological invariants of spaces

Typical Uses

Used to derive computable invariants and structural decomposition results for topological spaces.

Applications

  • Topological classification
  • Geometry and manifold theory
  • Applied topology pipelines

References

Recommended Textbooks

55M10 Dimension theory in algebraic topology

Overview

Dimension theory in algebraic topology. This topic contains classical constructions and results in algebraic topology, including foundational invariants and decomposition methods.

Related Wikipedia Page

Wikipedia: Algebraic topology

Useful Links

Key Ideas

  • classical homotopy and homology constructions
  • cellular and simplicial techniques
  • topological invariants of spaces

Typical Uses

Used to derive computable invariants and structural decomposition results for topological spaces.

Applications

  • Topological classification
  • Geometry and manifold theory
  • Applied topology pipelines

References

Recommended Textbooks

55M15 Absolute neighborhood retracts

Overview

Absolute neighborhood retracts. This topic contains classical constructions and results in algebraic topology, including foundational invariants and decomposition methods.

Related Wikipedia Page

Wikipedia: Algebraic topology

Useful Links

Key Ideas

  • classical homotopy and homology constructions
  • cellular and simplicial techniques
  • topological invariants of spaces

Typical Uses

Used to derive computable invariants and structural decomposition results for topological spaces.

Applications

  • Topological classification
  • Geometry and manifold theory
  • Applied topology pipelines

References

Recommended Textbooks

55M20 Fixed points and coincidences in algebraic topology

Overview

Fixed points and coincidences in algebraic topology. This topic contains classical constructions and results in algebraic topology, including foundational invariants and decomposition methods.

Related Wikipedia Page

Wikipedia: Algebraic topology

Useful Links

Key Ideas

  • classical homotopy and homology constructions
  • cellular and simplicial techniques
  • topological invariants of spaces

Typical Uses

Used to derive computable invariants and structural decomposition results for topological spaces.

Applications

  • Topological classification
  • Geometry and manifold theory
  • Applied topology pipelines

References

Recommended Textbooks

55M25 Degree, winding number

Overview

Degree, winding number. This topic contains classical constructions and results in algebraic topology, including foundational invariants and decomposition methods.

Related Wikipedia Page

Wikipedia: Algebraic topology

Useful Links

Key Ideas

  • classical homotopy and homology constructions
  • cellular and simplicial techniques
  • topological invariants of spaces

Typical Uses

Used to derive computable invariants and structural decomposition results for topological spaces.

Applications

  • Topological classification
  • Geometry and manifold theory
  • Applied topology pipelines

References

Recommended Textbooks

55M30 Ljusternik-Schnirelman (Lyusternik-Shnirel'man) category of a space

Overview

Ljusternik-Schnirelman (Lyusternik-Shnirel'man) category of a space. This topic contains classical constructions and results in algebraic topology, including foundational invariants and decomposition methods.

Related Wikipedia Page

Wikipedia: Algebraic topology

Useful Links

Key Ideas

  • classical homotopy and homology constructions
  • cellular and simplicial techniques
  • topological invariants of spaces

Typical Uses

Used to derive computable invariants and structural decomposition results for topological spaces.

Applications

  • Topological classification
  • Geometry and manifold theory
  • Applied topology pipelines

References

Recommended Textbooks

55M35 Finite groups of transformations in algebraic topology

Overview

Finite groups of transformations in algebraic topology. This topic contains classical constructions and results in algebraic topology, including foundational invariants and decomposition methods.

Related Wikipedia Page

Wikipedia: Algebraic topology

Useful Links

Key Ideas

  • classical homotopy and homology constructions
  • cellular and simplicial techniques
  • topological invariants of spaces

Typical Uses

Used to derive computable invariants and structural decomposition results for topological spaces.

Applications

  • Topological classification
  • Geometry and manifold theory
  • Applied topology pipelines

References

Recommended Textbooks