The construction of combinatorial manifolds with prescribed sets of links of vertices

A. A. Gaifullin

Izvestiya. Mathematics , Tome 72 (2008) no. 5, pp. 845-899

Résumé

To every oriented closed combinatorial manifold we assign the set (with repetitions) of isomorphism classes of links of its vertices. The resulting transformation $\mathcal{L}$ is the main object of study in this paper. We pose an inversion problem for $\mathcal{L}$ and show that this problem is closely related to Steenrod's problem on the realization of cycles and to the Rokhlin–Schwartz–Thom construction of combinatorial Pontryagin classes. We obtain a necessary condition for a set of isomorphism classes of combinatorial spheres to belong to the image of $\mathcal{L}$. (Sets satisfying this condition are said to be balanced.) We give an explicit construction showing that every balanced set of isomorphism classes of combinatorial spheres falls into the image of $\mathcal{L}$ 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. Given any singular simplicial cycle $\xi$ of a space $X$, this construction enables us to find explicitly a combinatorial manifold $M$ and a map $\varphi\colon M\to X$ such that $\varphi_*[M]=r[\xi]$ for some positive integer $r$. The construction is based on resolving singularities of $\xi$. We give applications of the 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.