TITLE: When does a CW-complex of dimension 2 embedd in $R^4$ ?
QUESTION [13 upvotes]: Let X be a finite CW-complex of dimension two having just one 0-cell
(+ finitely many 1-cells + finitely many 2-cells).
Is it true that X can be embedded in $R^4$ ?
If true, is it due to Stallings ?
More generally, let Y be a finite CW-complex of dimension two
of which the 1-skeleton embeds in $R^2$;
is is true that Y can be embedded in $R^4$ ?
(Reminder : any finite CW-complex of dimesnion two embeds in $R^5$.)
Any comment, or even better a reference, would help.

REPLY [2 votes]: Shapiro's obstruction:
A. Shapriro, "Obstructions to the imbedding of a complex in Euclidean space, I. The first obstruction," Ann. of Math., 66 No. 2 (1957), 256--269.