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