A Heegaard splitting of a connected orientable 3-manifold is any way of expressing as the union of two (3,1)-handlebodies along their boundaries. The boundary of such a (3,1)-handlebody is an orientable surface of some genus, which determines the number of handles in the (3,1)-handlebodies. Therefore, the handlebodies involved in a Heegaard splitting are the same, but they may be glued together in a strange way along their boundary. A diagram showing how the gluing is done is known as a Heegaard diagram.
Heegaard Splitting
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References
Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, p. 255, 1994.Referenced on Wolfram|Alpha
Heegaard SplittingCite this as:
Weisstein, Eric W. "Heegaard Splitting." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HeegaardSplitting.html