Talk:Supplement II (Ex)
Definition 0.1. Let be a subcomplex of a simplicial complex . The barycentric subdivision is the simplicial complex with -simplices for simplices of (totally-ordered by inclusion), and is the subcomplex whose -simplices are with for all . The supplement of in is the subcomplex whose -simplices are with for all .
Definition 0.2. For two spaces and , the join is by definition the quotient of by the equivalence relation given by
in other words is collapsed to at the end and is collapsed to at the end. So formally its elements are subject to this equivalence relation, but we will write them as formal sums . (This notation indicates that when it doesn't matter what is and when it doesn't matter what is.)
Construction 0.3. We will embed . First, we embed the subspaces and of in the obvious way, i.e.
Let be a simplex of which is not in or . So
with in and not in (). A point in is given in barycentric coordinates by
We define the embedding by the formula
These embeddings of , for , together with the embeddings of and above, glue to give a well-defined embedding of in . (Since the definitions agree on overlaps.)
Deformation retracts 0.4. Now define
and note that
There is then an obvious deformation retraction
where is homotopic to the identity via
Similarly there is a deformation retraction
with homotopic to the identity via