Simplicial join

Clearly, we have commutativity for arbitrary abstract simplicial complexes Ax and Z 2, the joins Ax A2 and Z 2 Ax are isomorphic. The join is also associative namely, for arbitrary abstract simplicial complexes Ax, A2, and As, the joins (Zi A2) As and Z i As As) are isomorphic. 

Another important property of the join is that for any abstract simplicial complex A and any simplex t A, the abstract simplicial complexes lk2i(r) r and star(r) are isomorphic. 

The join of an arbitrary abstract simplicial complex A with the empty simplex is equal to A. 

Most of the terminology, such as skeleton, subcomplex, join, carries over from the simplicial situation. One new property worth observing is that a direct product of two geometric polyhedral complexes is again a geometric polyhedral complex, whereas the same is not true for the geometric simplicial complexes. 

Corollary 20.6. For an arbitrary graph T, the simplicial complex Hom+(T,iC ) is isomorphic to the n-fold join of the independence complex ofT, which in our notation is called Ind(T) ". 

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