Sutured ECH and Contact 2-Handles

Abstract

We show that there are well-defined maps on sutured ECH induced by contact 2-handle attachments and that the sutured ECH contact class is functorial under such maps.

Figures (3)

• Figure 1: Left: $[-1 , 0] _ s \times A(\gamma) \simeq [-1 , 0] _ s \times [-1 , 1] \times \gamma$ in which the red arcs denote the part of the dividing curve on $\widetilde{u}(M)$. Right: After the $2$-handle is attached along $[-\tfrac{1}{4} , \tfrac{1}{4}] \times \gamma$ the part of $\partial \widetilde{u}(M) _ 2$ in this local picture topologically is a disjoint union of two disks which we call $D$ and $D'$ refers to the complement of $\{ 0 \} _ s \times ([ -1 , -\tfrac{3}{4}) \cup ( \tfrac{3}{4} , 1] ) \times \gamma$ in $D$.
• Figure 2:
• Figure 3: The upper red-shaded region denotes ${\partial _ h} ^ + E$ while the bottom red-shaded region denotes ${\partial _ h }^ - E$ upon which lie $a _ +$ and $a _ -$, respectively. The gray-shaded region denotes a collar neighborhood (in the $\tau$-direction) of $\partial _ v E$ upon which lies the collection of four arcs denoted by $a _ v$.

Theorems & Definitions (17)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Example 2.3
• Definition 2.4
• Theorem 2.5
• proof
• Corollary 2.6
• proof
• Proposition 3.1: cf. Section 4.2.3 in bs2