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In the paper Hardness of embedding simplicial complexes in $\mathbb{R}^{d}$ the abstract states that a finite simplicial complex of dimension $k$ embeds in $\mathbb{R}^{2k}$ while on page $856$ he says that it does not embed in $\mathbb{R}^{2k}$ but rather in $\mathbb{R}^{2k+1}$.

Which one is correct?

My understanding from the definition of a simplex $\Delta^k = \left\{(t_0,\cdots,t_k)\in\mathbb{R}^{k+1}\mid\Sigma_{i = 0}^{k}{t_i} = 1 \mbox{ and } t_i \ge 0 \mbox{ for all } i\right\}$

is that a finite simplicial complex of dimension $k$ embeds in $\mathbb{R}^{k+1}$, am I right?

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    @Sigur : so you are saying that after the gluing process we may lose the embeddebiliy in $\mathbb R^{k+1}$?2012-08-29

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