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On the disk complexes of weakly reducible, unstabilized Heegaard splittings of genus three III - Generalized Heegaard splittings and mapping classes | | Jungsoo Kim
; | | Date: |
1 Sep 2015 | | Abstract: | Let $M$ be an orientable, irreducible $3$-manifold admitting a weakly
reducible genus three Heegaard splitting as a minimal genus Heegaard splitting.
In this article, we prove that if $[f]$, $[g]in Mod(M)$ give the same
correspondence between two isotopy classes of generalized Heegaard splittings
consisting of two Heegaard splittings of genus two, say
$[mathbf{H}] o[mathbf{H}’]$, then there exists a representative $h$ of the
difference $[h]=[g]cdot[f]^{-1}$ such that (i) $h$ preserves a suitably chosen
embedding of the Heegaard surface $F’$ obtained by amalgamation from
$mathbf{H}’$ which is a representative of $[mathbf{H}’]$ and (ii) $h$ sends a
uniquely determined weak reducing pair $(V’,W’)$ of $F’$ into itself up to
isotopy. Moreover, for every orientation-preserving automorphism $ ilde{h}$
satisfying the previous conditions (i) and (ii), there exist two elements of
$Mod(M)$ giving correspondence $[mathbf{H}] o[mathbf{H}’]$ such that
$ ilde{h}$ belongs to the isotopy class of the difference between them. | | Source: | arXiv, 1509.0157 | | Services: | Forum | Review | PDF | Favorites | |
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