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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

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