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Thinning genus two Heegaard spines in \(S^ 3\). (English) Zbl 1048.57002
A Heegard spine of compact 3-manifold is a graph whose regular neighbourhood has as its closed complement a handlebody. One can construct such a spine by adding a number of arcs to a tunnel number one link. It is a well-known theorem of Waldhausen that any Heegard splitting of \(S^3\) can be isotoped to a standard one of the same genus. Suppose the trivalent graph \(\Gamma\) is an arbitrary Heegard spine of \(S^3\) (the notion of thin position is defined). Suppose a height function is given on \(S^3\) and we isotope \(\Gamma\) in \(S^3\) to make it as thin as possible. What can we say about the positioning of \(\Gamma\)? The authors answer the question for genus two Heegard spines: once such a spine is put in thin position, some simple edge (not a loop) will be level. The similar question for higher genus is still open.

MSC:
57M10 Covering spaces and low-dimensional topology
57N10 Topology of general \(3\)-manifolds (MSC2010)
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