I've heard that there always is an embedding in $R^7$ (can someone provide a reference for that?) and this number cannot be lowered in general. But I'm interested in a somewhat special case, namely: complexes that are small deformations of complexes embeddable in $R^3$. In other words - start with a simplical complex in $R^3$ and change the dimensions of some of the cells. The resulting complex in general cannot be embedded in $R^3$ any more and $R^7$ always suffices. Is $R^6$ enough in this case?
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2$\begingroup$ ${\bf R}^7$ is presumably a special case of the embedding of any finite $n$-complex in $2n+1$ dimensional space: first embed in a space of really high dimension by sending each vertex to a different basis vector; then project to a random ${\bf R}^{2n+1}$. This works because generically $n$-dimensional subspaces of $2n+1$ dimensional space don't meet. [And you might want to correct the typo in the title: missing the last I in "simplicIal".] $\endgroup$– Noam D. ElkiesCommented Jun 21, 2011 at 16:05
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2$\begingroup$ What do you mean by "change the dimensions of some of the cells" ? $\endgroup$– BS.Commented Jun 21, 2011 at 16:06
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$\begingroup$ @BS: I assume he means to change the sizes of the cells. $\endgroup$– Andreas BlassCommented Jun 21, 2011 at 18:45
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1 Answer
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There's an obstruction to embedding $n$-complexes in $\mathbb R^{2n}$ provided $n \geq 3$ due to Shapiro.
MR0089410 (19,671a) Shapiro, Arnold Obstructions to the imbedding of a complex in a euclidean space. I. The first obstruction. Ann. of Math. (2) 66 (1957), 256–269.
or as Ian suggests, van Kampen pre-dates Shapiro.
JFM 58.0615.03 van Kampen, E. R. Berichtung zu:``Komplexe in euklidischen Räumen''. (German) Abhandlungen Hamburg 9, 152-153 (1932).
Could you be more precise on how you want to modify the your complexes that originally are in $\mathbb R^3$, your special case?
answered Jun 21, 2011 at 16:08
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1$\begingroup$ Many people call this Van Kampen's embedding obstruction: springerlink.com/content/m71886j8mu666061 $\endgroup$– Ian AgolCommented Jun 21, 2011 at 16:56