Variable Dimension Complexes Part I Basic Theory

Cover Variable Dimension Complexes Part I Basic Theory
Variable Dimension Complexes Part I Basic Theory
Robert Michael Freund
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We say x is full X if |x| = |t I +1. X is a full simplex if it is a maximum-dimension simplex in A(T ) .
For each TfJ. We also define 3'A(T) as 8'A(T) = {x € 3A(T)1t = T}.
We illustrate the above definitions in the V-complex in Figure 2. In the fiflure, the left-most vertex of the 2-simplex is A(0), the "bottom" line segment is A(l), the left-sided line segment is A(2), and the simplex itself is A(l, 2).
For X = {d, e}, T = {1, 2}, for x = {f, g}, T = {1}. For X = {e, f, h}, T = {1, 2}. The sim
...plices {a}, {f, g}, and {e, h, f} are all full. We have 9'A(1) = {c}, 8'A(2) = {b}, and 8'A(1, 2) is the pseudomanifold corresponding to the line segment from b to c. Thus, while both {k, £, } and {f, g} are elements of 3A(1, 2), {k, Jl} (. 9'A(1, 2), whereas {f, g} ^ 3'A(1, 2).
For T = 0, A(T) contains only one vertex, the origin, and the empty set 0, and therefore 3'A(0) = {0}.
Figure 1(a) Af2) A0. 2) ^(2. 3) "MW -ACD- AC3) A(1, 3) Figure 1 (b^ A(4) A(6) Figure 1(c) - 11 - Let K be a V-complex with label set N.


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