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

Thurston, William P.

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

Keywords:

3-manifold; transversely orientable foliation; compact leaf; homology and cohomology norms

Citations:

Zbl 0533.57013; Zbl 0516.53046