On the $2$-realizability of $2$-types

Abstract:

A $2$-type is a triple $(\pi ,{\pi _2},k)$, where $\pi$ is a group, ${\pi _2}$ a $\pi$-module and $k \in {H^3}(\pi ,{\pi _2})$. The following question is studied: When is a $2$-type $(\pi ,{\pi _2},k)$ realizable by $2$-dimensional CW-complex $X$ such that the $2$-type $({\pi _1}X,{\pi _2}X,k(X))$ is equivalent to $(\pi ,{\pi _2},k)$? A long list of necessary conditions is given (2.2). One necessary and sufficient condition (3.1) is proved, provided $\pi$ has the property that stably free, finitely generated $\pi$-modules are free. “Stable” $2$-realizability is characterized (4.1) in terms of the Wall invariant of [15]. Finally, techniques of [5] are used to extend C. T. C. Wall’s Theorem F of [15] to a space $X$ which is dominated by a finite CW-complex of dimension 2, provided ${\pi _1}X$ is finite cyclic. Under these conditions $X$ has the homotopy type of a finite $2$-complex if and only if the Wall invariant vanishes.