Inherited asphericity, links and identities among relations

Abstract

AbstractIf a two-dimensional CW-complex is aspherical (i.e.π2 = 0), are its subcomplexes also aspherical? We approach this famous conjecture, originally presented by J.H.C. Whitehead, in a most direct fashion.The (finite) conjecture can easily be reduced to complexes with only one 0-cell and to complexes where the subcomplex has only one less2-cell than the parent complex. If an element of the second homotopy group of the subcomplex is given, it extends to a map from B3 to the larger complex. The inverse image of the midpoints of disks is a link together with a collection of closed intervals with endpoints on the boundary (after well positioning). Thinking of the disk which is not in the subcomplex as being green and the others as being red and coloring the inverse image accordingly (note the intervals are all red), we see that, if we knew enough about these links and the moves we can make on them, we might be able to make the green components go away; if we could do this for the general case, the conjecture would be proved