The Thurston norm and 2-handle addition. (English) Zbl 0627.57010

Define the complexity \(\chi_-(S)\) of an oriented surface S to be - \(\chi\) (C), where C is the union of all nonsimply connected components of S and \(\chi\) (C) its Euler characteristic. For M a compact oriented 3- manifold and N a (possibly empty) surface in \(\partial M\), assign to any homology class \(\alpha\) in \(H_ 2(M,N; {\mathbb{Z}})\) the minimum complexity v of all oriented embedded surfaces whose fundamental class represents \(\alpha\).
There is a unique continuous extension of v to \(H_ 2(M,N; {\mathbb{R}})\). It is a pseudonorm [W. P. Thurston, Mem. Am. Math. Soc. 339, 99-130 (1986; Zbl 0585.57006)]. Any continuous map \(\phi\) : (X,Y)\(\to (M,N)\) induces a pseudonorm \(\phi^*(v)\) on \(H_ 2(X,Y)\) defined by \(\phi^*(v)(\alpha)=v(\phi_*(\alpha))\). If \(\phi\) : (M’,N’)\(\to (M,N)\) is an inclusion of 3-manifolds, then \(\phi^*(v)\leq v'\). The following is the main result of the paper. This generalizes a series of results due to the reviewer, Jaco, and Johannson.
Theorem. Let \(M^+\) be the 3-manifold obtained from a compact oriented 3-manifold M by attaching a 2-handle along an annulus A in a (possibly empty) surface N in \(\partial M\). If N is compressible but N-A is not, then \(\phi^*(v^+)=v\).
Reviewer: J.Przytycki


57N10 Topology of general \(3\)-manifolds (MSC2010)
57M35 Dehn’s lemma, sphere theorem, loop theorem, asphericity (MSC2010)


Zbl 0585.57006
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