By Weyl H.

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Geometry of the simplicial join One can define the join of arbitrary topological spaces. Let I denote the closed unit interval. 31. Let X and Y be two topological spaces. The join of X and Y is the topological space X ∗ Y defined as follows: X ∗ Y = I × X × Y / ∼, where the equivalence relation ∼ is given by • • (0, x, y) ∼ (0, x, y˜), for all y, y˜ ∈ Y ; (1, x, y) ∼ (1, x ˜, y), for all x, x ˜ ∈ X. 9) where on the righthand side we take the simplicial join. Given geometric realizations of ∆1 in Rm and ∆2 in Rn , a geometric realization of ∆1 ∗ ∆2 in Rm+n+1 can be obtained as follows.

We start with a discrete set of points. This is our 0-skeleton, and we proceed by induction on the dimension of the attached faces. At step d we attach the d-dimensional faces, all at once. Each face is represented by some convex polytope P in the sample space Rd . To attach it we need a continuous map f : ∂P → X, where X denotes the part of the complex created in the first d − 1 steps. The attaching map must satisfy an additional condition: we request that it should be a homeomorphism between ∂P and f (∂P ), and that this homeomorphism should preserve the cell structures, where the cell structure on ∂P is simply the given polytopal structure, and the cell structure on f (∂P ) is induced from the previous gluing process.

K. We also remark that instead of taking the successive joins, one can think of X1 ∗ · · · ∗ Xk as the quotient space X1 ∗ · · · ∗ Xk = ∆[k] × X1 × · · · × Xk / ∼, where the equivalence relation ∼ is given by (α, x1 , . . , xk ) ∼ (α, x′1 , . . , x′k ) if tuples (x1 , . . , xk ) and (x′1 , . . , x′k ) coincide on the support simplex of α (where the support simplex of α is the minimal subsimplex of ∆[k] containing α). 22 2 Cell Complexes Geometry of barycentric subdivision The geometric realizations of the abstract simplicial complexes Bd ∆ and ∆ are related in a fundamental way.