Join |

**For Simplices**

$ __Def__: Given a point
*P* and an affine singular *q*-simplex *σ*
= (*P*_{0}*,
P*_{1}, ...,
*P*_{q}) in an affine
space, their join is the affine singular (*q*+1)-simplex

*P**σ*:=
(*P*, *P*_{0},
*P*_{1}, ...,
*P*_{q}) .

* __Idea__: The simplex obtained
by "joining *P* with all the vertices of *σ*".

* __Note__: *P* doesn't have to
be "outside *σ*", since these simplices are singular.

* __For chains__: If, instead of
a simplex *σ*, we have a singular *q*-chain *c*,
we define the join *Pc* by using linearity.

@ __References__: in Nash & Sen 83, p85.

**PL Join of Subspaces X, Y of R^{n}**

*

**For Topological Spaces X and Y**

$

(*x*, 1, *y*) ~ (*x'*, 1, *y*),
and
(*x*, 0, *y*) ~ (*x*, 0, *y'*) .

* __To visualize__: Consider
(*X* × *Y*) × I, and squeeze (*X* ×
*Y*) × {0} down to *Y*, and (*X* × *Y*)
× {1} down to *X*.

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send feedback and suggestions to bombelli at olemiss.edu – modified 27 nov 2011