Construction of combinatorial manifolds with the prescribed sets of links of vertices

TL;DR

This paper studies the inverse problem of reconstructing combinatorial manifolds from vertex link sets, providing explicit constructions and applications to cobordism theory and Pontryagin classes.

Contribution

It offers an explicit construction method for realizing balanced sets of vertex links as images of the transformation L, linking to classical topological problems.

Findings

Provides a construction for realizing balanced sets of links as images of L

Enables explicit construction of combinatorial manifolds from cycles

Proves that rational cobordism invariants have local formulas

Abstract

To each oriented closed combinatorial manifold we assign the set (with repetitions) of isomorphism classes of links of its vertices. The obtained transformation L is the main object of study of the present paper. We pose a problem on the inversion of the transformation L. We shall show that this problem is closely related to N.Steenrod's problem on realization of cycles and to the Rokhlin-Schwartz-Thom construction of combinatorial Pontryagin classes. It is easy to obtain a condition of balancing that is a necessary condition for a set of isomorphism classes of combinatorial spheres to belong to the image of the transformation L. In the present paper we give an explicit construction providing that each balanced set of isomorphism classes of combinatorial spheres gets into the image of L after passing to a multiple set and adding several pairs of the form (Z,-Z), where -Z is the sphere Z with the orientation reversed. This construction enables us, for a given singular simplicial cycle of a space R, to construct explicitly a combinatorial manifold M and a mapping $\phi:M\to R$ such that $\phi_*[M]=r[\xi]$ for some positive integer r. The construction is based on resolving singularities of the cycle $\xi$. We give applications of our main construction to cobordisms of manifolds with singularities and cobordisms of simple cells. In particular, we prove that every rational additive invariant of cobordisms of manifolds with singularities admits a local formula. Another application is the construction of explicit (though inefficient) local combinatorial formulae for polynomials in the rational Pontryagin classes of combinatorial manifolds.