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Séminaire Lotharingien de Combinatoire, 80B.75 (2018), 12 pp.

# Emily Barnard

# The Canonical Join Complex of the Tamari Lattice

**Abstract.**
In this paper, we study a simplicial complex on the elements of the
Tamari lattice in types *A* and *B* called the canonical join complex.
The canonical join representation of an element *w* in a lattice *L*
is the unique lowest expression V*A* for *w*.
We abuse notation and also say that the set *A* is a canonical join
representation (when we mean V*A* is a canonical join
representation).
The collection of all such subsets is an abstract simplicial complex
called the canonical join complex of *L*.
We realize the canonical join complex of the Tamari lattice as a
complex of noncrossing arc diagrams, give a shelling order on its
facets, and show that it is homotopy equivalent to a wedge of
Catalan-many spheres.

Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.

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