On J. H. C. Whitehead's aspherical question I

Abstract

A connected, finite two-dimensional CW-complex with fundamental group isomorphic toG is called a [G, 2] f -complex. LetL⊲G be a normal subgroup ofG. L has weightk if and only ifk is the smallest integer such that there exists {l1,…,lk}⊆L such thatL is the normal closure inG of {l1,…,lk}. We prove that a [G, 2] f -complexX may be embedded as a subcomplex of an aspherical complexY=X∪{e 21 ,…,e 2k } if and only ifG has a normal subgroupL of weightk such thatH=G/L is at most two-dimensional and defG=defH+k. Also, ifX is anon-aspherical [G, 2] f -subcomplex of an aspherical 2-complex, then there exists a non-trivial superperfect normal subgroupP such thatG/P has cohomological dimension ≤2. In this case, any torsion inG must be inP.