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An embedding for \(\pi_2\) of a subcomplex of a finite contractible two-complex. (English) Zbl 0736.57003

An algebraic conjecture, phrased in the language of combinatorial group theory and referred to as (ARWC), is shown to imply that any subcomplex of a finite contractible 2-complex is aspherical. Thus the truth of (ARWC) would settle an important special case of a well known problem of J. H. C. Whitehead [Ann. Math., II. Ser. 42, 409-428 (1941; Zbl 0027.26404)]. Moreover a somewhat weaker version of (ARWC) is established; the proof involves the free Lie algebra associated with a free group in the usual way.
{Reviewer’s remark: Related results and more references about Whitehead’s problem may be found in the reviewer’s paper [Math. Ann. 258, 17-37 (1981; Zbl 0458.57001)].}.

MSC:

57M20 Two-dimensional complexes (manifolds) (MSC2010)
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