# gt.geometric topology – Wanted: a nontrivial weakly inadmissible Heegaard diagram Answer

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## gt.geometric topology – Wanted: a nontrivial weakly inadmissible Heegaard diagram

If you may have any 3-manifold $$Y$$ with optimistic $$b_1$$, and any (universal) Heegaard diagram $$mathcal{H} = (Sigma, alpha, beta)$$ for it, you may place the basepoint $$z$$ in order that $$(Sigma, alpha, beta,z)$$ shouldn’t be admissible.

In truth, deem any province $$P_0$$ in $$mathcal{H}$$ whose border is a nontrivial sum of $$alpha$$– and $$beta$$-curves (this might breathe referred to as sporadic, if the habitual definition of periodicity did not require that the multiplicity of the basepoint breathe 0), and let $$m$$ breathe the minimal multiplicity of $$P_0$$, attained at a area $$D$$. Consider $$P = P_0 – mcdot Sigma$$, and place the basepoint $$z$$ in $$D$$. Clearly, $$P$$ is a nonzero sporadic province in $$(Sigma, alpha, beta,z)$$ which has solely non-negative multiplicities.

Here’s what occurs for the “standard” admissible-looking diagram for $$S^1times S^2$$ you advert to. (I’m linking to an SVG portray, that I am unable to metamorphose on the significance — if anybody desires to edit, be happy to do it). The “standard” sporadic province is the dissimilarity of the 2 bigonal areas. Adding the all torus and inserting the basepoint within the 0-multiplicity bigon yields the linked portray.
Colour signifies multiplicity: white means 0, lighter grey 1, darker grey 2.

EDIT: as identified by the OP in a observation under, the province above has index 2, whereas we would love to have an index-0 sporadic province. I cerebrate I’ve an instance for this phenomenon (once more, a diagram for $$S^2times S^1$$):

Consider the province $$P_0$$ proven within the portray under (yellow means multiplicity -1, white multiplicity 0, and grey multiplicity 1), and generator $$x$$ akin to the 2 thick black factors, and the inexperienced basepoint $$z$$. The index of $$P_0$$ as a province in $$pi_2(x,x)$$ is -2 (the Euler quantity is the Euler attribute of the doubly-pointed torus, which is -2), and the multiplicity at $$x$$ is zero. If you add the gross floor, you get a sporadic province of index 0 with solely non-negative multiplicities.

Notice that there’s a lot of redundancy, each within the variety of turbines and the genus of the diagram, however I would not plane attempt to formulate a query the place that is addressed.

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