“School of Mathematics”
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| Paper IPM / M / 17088 |
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| Abstract: | |
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We say a d-dimensional simplicial complex embeds into double
dimension if it embeds into the Euclidean space of dimension 2d. For instance,
a graph is planar iff it embeds into double dimension. We study the conditions
under which the join of two simplicial complexes embeds into double dimension.
Quite unexpectedly, we show that there exist complexes which do not
embed into double dimension, however their join embeds into the respective
double dimension. We further derive conditions, in terms of the van Kampen
obstructions of the two complexes, under which the join will not be embeddable
into the double dimension.
Our main tool in this study is the definition of the van Kampen obstruction
as a Smith class. We determine the Smith classes of the join of two Zpcomplexes
in terms of the Smith classes of the factors. We show that in
general the Smith index is not stable under joins. This allows us to prove our
embeddability results.
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