Universal simplicial complexes inspired by toric topology

Abstract

Let k be the field Fp or the ring Z. We study combinatorial and topological properties of the universal complexes X(k n ) and K(k n ) whose simplices are certain unimodular subsets of k n . We calculate their f -vectors, show that they are shellable but not shifted, and find their applications in toric topology and number theory.Using discrete Morse theory, we detect that X(k n ), K(k n ) and the links of their simplicies are homotopy equivalent to a wedge of spheres specifying the exact number of spheres in the corresponding wedge decompositions. This is a generalisation of Davis and Januszkiewicz's result that K(Z n ) and K(F n 2 ) are (n − 2)-connected simplicial complexes.

Equivariant Bordism of 2-Torus Manifolds and Unitary Toric Manifolds

“…n ) can be regarded as a chian complex with chain map d * . This point of view is similar to the definition of universal Z n -space X(Z n ) ( [1]). Definition 3.6.…”

“…The collection of all unimodular subsets of Z n 2 forms a simplicial complex X(Z n 2 ) on unimodular vertices, which is called the universal Z n 2 -complex. As showed in [1], the universal complex X(Z n…”

“…It is clear that the simplicial complexes X(Z n ) is (n − 1)-dimensional and pure. As showed in [1], the universal complex X(Z n…”

“…The homotopy of the universal complex X(Z n 2 ) was determined in [1], we recall their result to calculate the dimension of Z n (Z n 2 ), the proof is straightforward. Theorem 2.6.…”

“…The homotopy type of X(Z n ) was proved to be the wedge of countably infinite spheres S n−1 (see [1]). So for n ≥ 2, we have Z U 2 (T 1 ) ∼ = Z and Z U 2n (T n ) is isomorphic to a countably infinite direct sum of Z (Corollary 3.8).…”