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The Browder-Livesay invariants provide obstructions to the realization of elements of Wall groups by normal maps of closed manifolds. A generalization of the iterated Browder-Livesay invariants is proposed and properties of the invariants obtained are described. The generalized definition makes it possible to investigate the relationship between a normal map and its restriction to a submanifold and clarifies the relationship between the Browder-Livesay invariants and the Browder-Quinn groups of obstructions to surgery on filtered manifolds. Several theorems describing a relationship between a normal map and its restriction to a submanifold are proved. [ABSTRACT FROM AUTHOR]

We consider the class G4 of Morse—Smale diffeomorphisms on $$ \mathbb{S} $$3 with nonwandering set consisting of four fixed points (namely, one saddle, two sinks, and one source). According to Pixton, this class contains a diffeomorphism that does not have an energy function, i.e., a Lyapunov function whose set of critical points coincides with the set of periodic points of the diffeomorphism itself. We define a quasi-energy function for any Morse—Smale diffeomorphism as a Lyapunov function with the least number of critical points. Next, we single out the class G4,1 ⊂ G4 of diffeomorphisms inducing a special Heegaard splitting of genus 1 of the sphere $$ \mathbb{S} $$3. For each diffeomorphism in G4,1, we present a quasi-energy function with six critical points. [ABSTRACT FROM AUTHOR]

In 1991, Turaev and Viro constructed a quantum topological linear representation of mapping class groups of closed surfaces. To the mappings of a surface into itself, they assigned simple polyhedra whose boundaries consisted of two simple graphs cutting the surface into cells. The computational complexity of the Turaev-Viro representations strongly depends on the choice of suitable sets of simple polyhedra. In this paper, simple polyhedra for the torus are constructed. One of the reasons why they are convenient is that they all are obtained by gluing along boundary of copies of the same simple polyhedron. [ABSTRACT FROM AUTHOR]