We prove that if a finite lattice L has order dimension at most d, then the homology of the order complex of its proper part Lo vanishes in dimensions d - 1 and higher. If L can be embedded as a join-sublattice in Nd, then Lo actually has the homotopy type of a simplicial complex with d vertices.
Bibliographical noteFunding Information:
The authors thank Irena Peeva for the initial conversations leading to this work and to , and the Mathematisches Forschungsinstitut Oberwolfach for their hospitality. They also thank Günter M. Ziegler for helpful conversations and edits. Vic Reiner was supported by a University of Minnesota McKnight-Land Grant Fellowship and a Sloan Fellowship, and Volkmar Welker was supported by Deutsche Forschungsgemeinschaft (DFG).
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- Order complex
- Order dimension
- Topology of posets