BooksStandalone TitlesCollected Volumepp. 315–330

Combinatorial realisation of cycles and small covers

  • Alexander A. Gaifullin

    Steklov Mathematical Institute, Moscow, Russian Federation
Combinatorial realisation of cycles and small covers cover
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In 1940s Steenrod asked if every homology class zHn(X;Z)z \in H_n(X; \mathbb Z) of every topological space XX can be realised by an image of the fundamental class of an oriented closed smooth manifold. Thom found a non-realisable 7-dimensional class and proved that for every nn, there is a positive integer k(n)k(n) such that the class k(n)zk(n)z is always realisable. The proof was by methods of algebraic topology and gave no information on the topology of the manifold which realises the homology class. We give a purely combinatorial construction of a manifold that realises a multiple of a given homology class. For every nn, this construction yields a manifold M0nM^n_0 with the following universality property: For any XX and zHn(X;Z)z \in H_n(X; \mathbb Z), a multiple of zz can be realised by an image of a (non-rami ed) finite-sheeted covering of M0nM^n_0 . Manifolds satisfying this property are called URC-manifolds. The manifold M0nM^n_0 is a so-called small cover of the permutahedron, i. e., a manifold glued in a special way out of 2n2n permutahedra. (The permutahedron is a special convex polytope with (n+1)(n+1)! vertices.) Among small covers over other simple polytopes, we find a broad class of examples of URC-manifolds. In particular, in dimension 4, we fi nd a hyperbolic URC-manifold. Thus we prove a conjecture of Kotschick and Loh claiming that a multiple of every homology class can be realised by an image of a hyperbolic manifold. We also obtain some new results on the relationship of URC-manifolds with theory of simplicial volume.