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Topological structure of $$k$$-saddle surfaces. (English) Zbl 0824.53006
Bokut’, L. A. (ed.) et al., Third Siberian school on algebra and analysis. Proceedings of the third Siberian school, Irkutsk State University, Irkutsk, Russia, August 30-September 4, 1989. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 163, 29-33 (1995).
A complete $$n$$-dimensional submanifold $$F^ n$$ in the Euclidean space $$E^ m$$ is called a $$k$$-saddle surface if $$H_ k(F^ n, F^ n \cap E^ r) \cong 0$$ for any $$r$$-dimensional Vietoris integral homology group. The class of $$n$$-dimensional $$k$$-saddle surfaces of class $$C^ 3$$ is denoted by $${U_ k}^ n$$.
The author investigates the topological structure of $$k$$-saddle surfaces. In particular, he proves that for any $$F^ n \in {U_ k}^ n$$, $$n \geq 6$$, $$k \leq n-2$$, there exists a locally tame compact $$(k-1)$$-dimensional polyhedron which is a strong deformation retract of $$F^ n$$ if and only if $$F^ n$$ is homeomorphic to the interior of a compact manifold with boundary. Furthermore, this condition is verified if the saddle surface is simply connected and its homology groups are finitely generated.
For the entire collection see [Zbl 0816.00016].
##### MSC:
 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces 53C40 Global submanifolds