In this talk, we will discuss some open problems in low-dimensional topology, including Wall’s D(2) problem, normal generation conjecture (the Wiegold Conjecture) of perfect groups, Swan’s problem on partial Euler characteristic and deficiency of groups. We will then present a relationship among these problems. As applications, we will prove that for a 3-dimensional complex X of cohomological dimension 2 with a finite fundamental group, assuming the Wiegold conjecture holds, the complex X is homotopy equivalent to a finite 2-complex after wedging a copy of sphere S^2.  If the time allowed, we will also discuss an application to the Whitehead asphericity conjecture. This is a joint work with Feng Ji.

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