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Average Euler characteristic of leaves of codimension-one foliations. (English) Zbl 0658.57014
Foliations, Proc. Symp., Tokyo 1983, Adv. Stud. Pure Math. 5, 395-415 (1985).
[For the entire collection see Zbl 0627.00017.]
As an invariant of the quasi-isometry type of a non-compact 2-dimensional manifold, A. Phillips and D. Sullivan [Topology 20, 209-218 (1981; Zbl 0454.57016)] introduced the notion of having average Euler characteristic zero. To generalize the notion to the case of a higher dimensional manifold some modifications are necessary. Author’s definition: A non-compact Riemannian manifold (F,g) (dim F$$>2)$$ has average Euler characteristic zero if (F,g) has a hereditarily uniform triangulation T and a sequence of subcomplexes of T, $$F_ 1\subset F_ 2\subset...\subset F$$, which are compact connected p.l. manifolds such that (1) $$\{F_ i\}$$ is comparable to metric balls $$\{D_ r(x)\}$$, $$r\in {\mathbb{R}}_+$$ for some $$x\in F$$, (2) $$\lim_{i\to \infty}vol(\partial F_ i)/vol(F_ i)=0$$, (3) $$\lim_{i\to \infty}\chi (F_ i)/vol(F_ i)=0$$ $$(\chi (F_ i)=usual$$ Euler characteristic). Here, hereditary uniformity is a certain uniformity condition of simplices of T and its subdivisions. Among other things, the author proves the following: Theorem. Let M be a closed orientable manifold, $${\mathcal F}^ a$$transversely orientable codimension one smooth foliation of M and F a non-compact leaf of $${\mathcal F}$$. Suppose that (1) $$H_ 1(M;{\mathbb{R}})=0$$ if dim F$$=even$$, (2) F has non-exponential growth. Then F has average Euler characteristic zero. This theorem corresponds to the theorem of Phillips and Sullivan, which has the same conclusion in the case of a 2-dimensional leaf of a foliation of M such that $$H_ 2(M;{\mathbb{R}})=0$$.
Reviewer: T.Mizutani

##### MSC:
 57R30 Foliations in differential topology; geometric theory