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.

57Sxx Topological transformation groups

This subtopic introduces the core ideas in topological transformation groups, 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

57S05 Topological properties of groups of homeomorphisms or diffeomorphisms

Overview

Topological properties of groups of homeomorphisms or diffeomorphisms. This topic studies topological transformation groups and actions on manifolds and spaces, including orbit structures and equivariant phenomena.

Related Wikipedia Page

Topological group action (Wikipedia)

Useful Links

Key Ideas

  • group actions and orbit spaces
  • equivariant invariants
  • fixed-point and symmetry methods

Typical Uses

Used to exploit symmetry in topological problems and classify spaces with prescribed action behavior.

Applications

  • Equivariant topology
  • Symmetry in dynamics
  • Transformation geometry

References

Recommended Textbooks

57S10 Compact groups of homeomorphisms of topological manifolds

Overview

Compact groups of homeomorphisms of topological manifolds. This topic studies topological transformation groups and actions on manifolds and spaces, including orbit structures and equivariant phenomena.

Related Wikipedia Page

Topological group action (Wikipedia)

Useful Links

Key Ideas

  • group actions and orbit spaces
  • equivariant invariants
  • fixed-point and symmetry methods

Typical Uses

Used to exploit symmetry in topological problems and classify spaces with prescribed action behavior.

Applications

  • Equivariant topology
  • Symmetry in dynamics
  • Transformation geometry

References

Recommended Textbooks

57S12 Torus actions on manifolds

Overview

Torus actions on manifolds. This topic studies topological transformation groups and actions on manifolds and spaces, including orbit structures and equivariant phenomena.

Related Wikipedia Page

Topological group action (Wikipedia)

Useful Links

Key Ideas

  • group actions and orbit spaces
  • equivariant invariants
  • fixed-point and symmetry methods

Typical Uses

Used to exploit symmetry in topological problems and classify spaces with prescribed action behavior.

Applications

  • Equivariant topology
  • Symmetry in dynamics
  • Transformation geometry

References

Recommended Textbooks

57S15 Compact Lie groups of differentiable transformations

Overview

Compact Lie groups of differentiable transformations. This topic studies topological transformation groups and actions on manifolds and spaces, including orbit structures and equivariant phenomena.

Related Wikipedia Page

Topological group action (Wikipedia)

Useful Links

Key Ideas

  • group actions and orbit spaces
  • equivariant invariants
  • fixed-point and symmetry methods

Typical Uses

Used to exploit symmetry in topological problems and classify spaces with prescribed action behavior.

Applications

  • Equivariant topology
  • Symmetry in dynamics
  • Transformation geometry

References

Recommended Textbooks

57S17 finite transformation groups

Overview

finite transformation groups. This topic studies topological transformation groups and actions on manifolds and spaces, including orbit structures and equivariant phenomena.

Related Wikipedia Page

Topological group action (Wikipedia)

Useful Links

Key Ideas

  • group actions and orbit spaces
  • equivariant invariants
  • fixed-point and symmetry methods

Typical Uses

Used to exploit symmetry in topological problems and classify spaces with prescribed action behavior.

Applications

  • Equivariant topology
  • Symmetry in dynamics
  • Transformation geometry

References

Recommended Textbooks

57S20 Noncompact Lie groups of transformations

Overview

Noncompact Lie groups of transformations. This topic studies topological transformation groups and actions on manifolds and spaces, including orbit structures and equivariant phenomena.

Related Wikipedia Page

Topological group action (Wikipedia)

Useful Links

Key Ideas

  • group actions and orbit spaces
  • equivariant invariants
  • fixed-point and symmetry methods

Typical Uses

Used to exploit symmetry in topological problems and classify spaces with prescribed action behavior.

Applications

  • Equivariant topology
  • Symmetry in dynamics
  • Transformation geometry

References

Recommended Textbooks

57S25 Groups acting on specific manifolds

Overview

Groups acting on specific manifolds. This topic studies topological transformation groups and actions on manifolds and spaces, including orbit structures and equivariant phenomena.

Related Wikipedia Page

Topological group action (Wikipedia)

Useful Links

Key Ideas

  • group actions and orbit spaces
  • equivariant invariants
  • fixed-point and symmetry methods

Typical Uses

Used to exploit symmetry in topological problems and classify spaces with prescribed action behavior.

Applications

  • Equivariant topology
  • Symmetry in dynamics
  • Transformation geometry

References

Recommended Textbooks

57S30 Discontinuous groups of transformations

Overview

Discontinuous groups of transformations. This topic studies topological transformation groups and actions on manifolds and spaces, including orbit structures and equivariant phenomena.

Related Wikipedia Page

Topological group action (Wikipedia)

Useful Links

Key Ideas

  • group actions and orbit spaces
  • equivariant invariants
  • fixed-point and symmetry methods

Typical Uses

Used to exploit symmetry in topological problems and classify spaces with prescribed action behavior.

Applications

  • Equivariant topology
  • Symmetry in dynamics
  • Transformation geometry

References

Recommended Textbooks

57S99 None of the above

Overview

None of the above. This topic studies topological transformation groups and actions on manifolds and spaces, including orbit structures and equivariant phenomena.

Related Wikipedia Page

Topological group action (Wikipedia)

Useful Links

Key Ideas

  • group actions and orbit spaces
  • equivariant invariants
  • fixed-point and symmetry methods

Typical Uses

Used to exploit symmetry in topological problems and classify spaces with prescribed action behavior.

Applications

  • Equivariant topology
  • Symmetry in dynamics
  • Transformation geometry

References

Recommended Textbooks