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What is the Morse number of the Poincaré homology sphere? What about the stable Morse number?

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It's 6: you need one critical point of index 0 and index 3, and two of index 1 and index 2.

It's at least 6, since the only 3-manifolds with lower Morse number are the 3-sphere (2) and lens spaces (4).

It's at most 6, since there it has Heegaard genus at most 2. This is true for all 3-manifolds obtained as surgery along a 2-bridge knot: one place I know this is described is this paper by Jake Rasmussen.

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    $\begingroup$ A genus 2 Heegaard splitting for the Poincaré homology sphere is also provided in Rolfsen's "Knots and Links" section 9D. $\endgroup$ – Neil Hoffman Apr 9 '15 at 5:52

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