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-Kirby Calculus is a fundamental tool in [geometric topology](/wiki/Geometric_Topology), developed by Rob Kirby, for describing and manipulating 3- and 4-dimensional manifolds. It elegantly uses framed links in the 3-sphere to represent [surgery theory](/wiki/Surgery_Theory) operations, offering a visual and combinatorial approach to higher-dimensional spaces. This calculus provides a powerful framework for understanding manifold structure and invariants.
-- [Differential Topology](/wiki/Differential_Topology)
+Kirby Calculus is a fundamental tool in [geometric topology](/wiki/Geometric_Topology), developed by Rob Kirby, for describing and manipulating 3- and 4-dimensional [manifolds](/wiki/Manifold). It elegantly uses framed links embedded in the 3-sphere to represent [surgery theory](/wiki/Surgery_Theory) operations, offering a visual and combinatorial approach to understanding higher-dimensional spaces. This calculus provides a powerful framework for understanding manifold structure and invariants, particularly for the classification of smooth 4-manifolds.
+The core idea behind Kirby Calculus is that any closed, oriented 3-manifold can be obtained by integer [Dehn Surgery](/wiki/Dehn_Surgery) on a framed link in the 3-sphere. Moreover, any closed, oriented 4-manifold can be constructed by attaching 2-handles to the 4-ball along a framed link in its boundary 3-sphere. This connection allows topologists to translate complex questions about manifold structure into more tractable combinatorial problems involving framed links.
+A key aspect of the calculus is a set of "Kirby moves" (also known as Kirby transformations or Kirby calculus moves). These are local transformations on framed links that preserve the diffeomorphism type of the 4-manifold (or homeomorphism type of the 3-manifold) represented by the link. These moves allow topologists to simplify link diagrams, prove equivalences between different descriptions of the same manifold, and compute invariants.
+- [Surgery Theory](/wiki/Surgery_Theory)
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+Kirby Calculus is a fundamental tool in [geometric topology](/wiki/Geometric_Topology), developed by Rob Kirby, for describing and manipulating 3- and 4-dimensional manifolds. It elegantly uses framed links in the 3-sphere to represent [surgery theory](/wiki/Surgery_Theory) operations, offering a visual and combinatorial approach to higher-dimensional spaces. This calculus provides a powerful framework for understanding manifold structure and invariants.
+## See also
+- [Manifold](/wiki/Manifold)
+- [Knots](/wiki/Knots)
+- [Handle Decomposition](/wiki/Handle_Decomposition)
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