Universal realisators for homology classes

Abstract

Abstract. We study oriented closed manifolds M n possessing the following Universal Realisation of Cycles (URC ) Property: For each topological space X and each homology class z ∈ H n (X, Z), there exist a finite-sheeted covering M n → M n and a continuous mapping f : M n → X such that f * [ M n ] = kz for a non-zero integer k. We find wide class of examples of such manifolds M n among so-called small covers of simple polytopes. In particular, we find 4-dimensional hyperbolic manifolds possessing the URC property. As a consequence, we obtain that for each 4-dimensional oriented closed manifold N 4 , there exists a mapping of non-zero degree of a hyperbolic manifold M 4 to N 4 . This was conjectured by Kotschick and Löh.

Small covers of graph-associahedra and realization of cycles

“…In this section we describe an auxiliary combinatorial construction that will be used in the next section in the proof of Theorem 2.5. This construction generalizes the construction suggested by the author in [19] and developed in [20], [21], [22], and [23].…”

“…The problem arises: Which other manifolds M n , except for the Tomei manifold M n 0 , satisfy the same universal property with respect to the problem on realization of cycles? The author [23], [24] has introduced the following class of URC-manifolds (form the words "Universal Realizator of Cycles"). Definition 1.1.…”

“…(Strict definitions of a small cover and of the permutahedron will be given in Section 2.) In [23] the author has found many other examples of URC-manifolds among small covers of other simple polytopes. However, all these simple polytopes, except for the dodecahedron in dimension 3, are more complicated than the permutahedron.…”

“…We shall present an explicit construction that, for a singular cycle representing the given homology class, builds a manifold being a finite-fold covering of M P Γ ,λ that realizes a multiple of the given homology class. This construction is a further modification of the construction suggested by the author in [19] and developed in [20], [21], and [23]. Notice that another modification of this construction was applied to solve the problem of constructing of an oriented simplicial manifold with the prescribed set of links of vertices, see [22].…”

Buchstaber invariant theory of simplicial complexes and convex polytopes

“…For example, Buchstaber, Panov, and Ray [11,34] showed that in dimensions greater than 2, every class of complex cobordisms contains a connected quasitoric manifold with a natural stably complex structure compatible with the torus action. Gaifullin [14,40] showed that every n-dimensional homology class of a topological space X is realized with some multiplicity by the image of the fundamental cycle of the manifold that is a finite-sheeted covering of a small cover over a polytope P n , where one can take a sufficiently wide class of polytopes as P n . This class contains permutohedra [14] as well as all simple polytopes for which the boundary of the polar polytope is isomorphic to the barycentric subdivision of a simplicial sphere [40,Theorem 3.2].…”