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This subtopic introduces the core ideas in low-dimensional 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.
Knot theory. This topic studies low-dimensional topology, especially dimensions two through four, with strong interactions among manifolds, knots, and group actions.
Wikipedia: Low-dimensional topology
Used to classify and distinguish manifolds and embedded objects in dimensions where geometry and topology strongly interact.
Generalized knots (virtual knots, welded knots, etc.). This topic studies low-dimensional topology, especially dimensions two through four, with strong interactions among manifolds, knots, and group actions.
Wikipedia: Low-dimensional topology
Used to classify and distinguish manifolds and embedded objects in dimensions where geometry and topology strongly interact.
Knot polynomials. This topic studies low-dimensional topology, especially dimensions two through four, with strong interactions among manifolds, knots, and group actions.
Wikipedia: Low-dimensional topology
Used to classify and distinguish manifolds and embedded objects in dimensions where geometry and topology strongly interact.
Finite-type invariants of knots and 3-manifolds. This topic studies low-dimensional topology, especially dimensions two through four, with strong interactions among manifolds, knots, and group actions.
Wikipedia: Low-dimensional topology
Used to classify and distinguish manifolds and embedded objects in dimensions where geometry and topology strongly interact.
Homology theories for knots and links. This topic studies low-dimensional topology, especially dimensions two through four, with strong interactions among manifolds, knots, and group actions.
Wikipedia: Low-dimensional topology
Used to classify and distinguish manifolds and embedded objects in dimensions where geometry and topology strongly interact.
$2$-dimensional topology (including mapping class groups, Teichmüller theory, quantum groups). This topic studies low-dimensional topology, especially dimensions two through four, with strong interactions among manifolds, knots, and group actions.
Wikipedia: Low-dimensional topology
Used to classify and distinguish manifolds and embedded objects in dimensions where geometry and topology strongly interact.
General topology of $3$-manifolds. This topic studies low-dimensional topology, especially dimensions two through four, with strong interactions among manifolds, knots, and group actions.
Wikipedia: Low-dimensional topology
Used to classify and distinguish manifolds and embedded objects in dimensions where geometry and topology strongly interact.
Invariants of $3$-manifolds. This topic studies low-dimensional topology, especially dimensions two through four, with strong interactions among manifolds, knots, and group actions.
Wikipedia: Low-dimensional topology
Used to classify and distinguish manifolds and embedded objects in dimensions where geometry and topology strongly interact.
Hyperbolic $3$-manifolds. This topic studies low-dimensional topology, especially dimensions two through four, with strong interactions among manifolds, knots, and group actions.
Wikipedia: Low-dimensional topology
Used to classify and distinguish manifolds and embedded objects in dimensions where geometry and topology strongly interact.
Contact structures in $3$-manifolds. This topic studies low-dimensional topology, especially dimensions two through four, with strong interactions among manifolds, knots, and group actions.
Wikipedia: Low-dimensional topology
Used to classify and distinguish manifolds and embedded objects in dimensions where geometry and topology strongly interact.
Other geometric structures on $3$-manifolds. This topic studies low-dimensional topology, especially dimensions two through four, with strong interactions among manifolds, knots, and group actions.
Wikipedia: Low-dimensional topology
Used to classify and distinguish manifolds and embedded objects in dimensions where geometry and topology strongly interact.
General topology of $4$-manifolds. This topic studies low-dimensional topology, especially dimensions two through four, with strong interactions among manifolds, knots, and group actions.
Wikipedia: Low-dimensional topology
Used to classify and distinguish manifolds and embedded objects in dimensions where geometry and topology strongly interact.
Invariants of $4$-manifolds. This topic studies low-dimensional topology, especially dimensions two through four, with strong interactions among manifolds, knots, and group actions.
Wikipedia: Low-dimensional topology
Used to classify and distinguish manifolds and embedded objects in dimensions where geometry and topology strongly interact.
Symplectic structures in $4$-manifolds. This topic studies low-dimensional topology, especially dimensions two through four, with strong interactions among manifolds, knots, and group actions.
Wikipedia: Low-dimensional topology
Used to classify and distinguish manifolds and embedded objects in dimensions where geometry and topology strongly interact.
Other geometric structures on $4$-manifolds. This topic studies low-dimensional topology, especially dimensions two through four, with strong interactions among manifolds, knots, and group actions.
Wikipedia: Low-dimensional topology
Used to classify and distinguish manifolds and embedded objects in dimensions where geometry and topology strongly interact.