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By Weyl H.

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Low Dimensional Topology

During this quantity, that is devoted to H. Seifert, are papers in accordance with talks given on the Isle of Thorns convention on low dimensional topology held in 1982.

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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.