We announce that a minimal equivarient spine W for Teichmueller space Teich(S) consists of hyperbolic surfaces which are homologically filled by their shortest essential nonseparating geodesics. We thank MFB for comments on a previous proposal. This spine W is distinct from Thurston's proposal Th.
Our proposal obtains the optimal codimension, and we are amazed to see our construction finally explain the apparent numerical coincidence between dim H_1(S) -1 = 2g-1 and the precise codimension of the spine as predicted by the group cohomological formula dim Teich(S) - vcd MCG(S) = 2g-1, as demonstrated by Broaddus-Harer.
Our retract has some notable ingredients.
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We modify Thurston's original proposal and emphasize essential nonseparating geodesic curves in the surface. Thus we focus exclusively on essential nonseparating curves.
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We also emphasize collections of curves which homologically fill the surface. This is distinct from Thurston's original proposal that the spine consists of surfaces which are filled (in the standard sense) by their systoles. Indeed we were initially confused on this point, but discussions with MFB persuaded us that homological filling is the correct idea.
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We directly use Teichmueller's original idea that quadratic holomorphic differentials are infinitesimal deformations of hyperbolic structure, and we basically square harmonic one forms to obtain the quadratic holomorphic. The construction of the canonical harmonic one form is interesting variational lemma which depends on the curves not homologically filling.
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In Belt Tightening Lemma we prove that the lengths of curves are simultaneously increased by Teichmueller deformations in the direction of the canonical harmonic one forms constructed above. This is a direct computation of the lengths and readily follows from the construction of the harmonic one form.
[-JHM]