## A norm for the homology of 3-manifolds.(English)Zbl 0585.57006

Mem. Am. Math. Soc. 339, 99-130 (1986).
The paper is a slightly revised version of a preprint written 1976. For each compact, orientable 3-manifold M the author considers the function $$x: (M;{\mathbb{Z}})\to {\mathbb{Z}}$$ where $$x(a)=\inf \{\chi_-(S)$$, S is an embedded surface in M representing $$a\}$$, $$\chi_-(S)=\max \{0,-\chi (S)\}.$$ This function uniquely extends to a continuous linear on rays function $$x: H_ 2(M;{\mathbb{R}})\to {\mathbb{R}}$$ which is shown to be a pseudonorm, determined by a finite set of cohomology classes $$b_ 1,...,b_ k\in H^ 2(M;{\mathbb{Z}}),$$ namely: $$x(a)=\sup | <b_ i,a>|.$$ An analogous pseudonorm is introduced in $$H_ 2(M,\partial M;{\mathbb{R}}).$$ These pseudonorms are related to foliations in M.
In particular: If H is a 2-dimensional transversely orientable foliation in M such that F is transverse to $$\partial M$$ and F and $$F|_{\partial M}$$ have no Reeb components then every compact leaf of F is proven to attain the minimal $$\chi_-$$ in its homology class. Finally, a complete computation of x is given in a representative family of examples and 3 conjectures are stated. The first two conjectures were proven true by D. Gabai [J. Differ. Geom. 18, 445-503 (1983; Zbl 0533.57013)]. Note also M. Gromov’s paper [Publ. Math., Inst. Hautes Étud. Sci. 56, 5-99 (1982; Zbl 0516.53046)] which develops a more general theory of homology and cohomology norms.
For the entire collection see [Zbl 1415.57001].
Reviewer: V.Turaev

### MSC:

 57N10 Topology of general $$3$$-manifolds (MSC2010) 55N10 Singular homology and cohomology theory 57R30 Foliations in differential topology; geometric theory

### Citations:

Zbl 0533.57013; Zbl 0516.53046