= Admissible cut for a Heegaard Floer mixed map
{c}
An admissible cut splits a four-dimensional <cobordism> as $W=W_1\cup_N W_2$, with $b_2^+(W_1),b_2^+(W_2)>0$, and with the Mayer-Vietoris connecting map $H^1(N;\mathbb Z)\to H^2(W;\mathbb Z)$ equal to zero. The last condition removes the ambiguity in gluing <Spin-c structures> on the two pieces. Positivity makes both infinity cobordism maps vanish, allowing passage through the <Reduced Heegaard Floer homology> of $N$. A useful cut is the boundary of a neighborhood of a positive-square embedded surface, chosen so that its complement still has a positive direction. The nonzero normal Euler number makes $H^1$ of the neighborhood surject onto $H^1$ of its boundary, proving the connecting-map condition.
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