Coverings preserve the higher-homotopy obstruction to K(pi,1)

Fix $n\ge 2$. The cube $I^n$ is path-connected and locally path-connected, and every loop in it contracts to its basepoint by straight-line interpolation inside the cube. Thus its fundamental group is trivial. A representative $a:(I^n,\mathrm{\partial }{I}^n)\to (B,b)$ has a lift taking a chosen boundary point to $e$ by [F1]. The boundary $\mathrm{\partial }{I}^n$ is connected for $n\ge 2$; the lift maps it continuously into the discrete fibre $p^{-1}(b)$, so the entire boundary maps to $e$. Hence every class in ${\pi }_n(B,b)$ is in the image of $p_{\ast }$.

Let $X$ be either $E$ or $B$ and suppose that $X$ is of CW homotopy type. Choose a CW complex $C$, a homotopy equivalence $f:C\to X$ and an inverse $g:X\to C$, with homotopies $gf\simeq {\mathrm{id}}_C$ and $fg\simeq {\mathrm{id}}_X$. Since $X$ is nonempty, so is $C$. For any $c,d\in C$, a path in $X$ from $f(c)$ to $f(d)$ maps under $g$ to a path from $gf(c)$ to $gf(d)$; the first homotopy supplies paths from $gf(c)$ to $c$ and from $gf(d)$ to $d$. Reversing the first and concatenating shows that $C$ is path-connected.

Suppose two based cube maps into $E$ have images that are homotopic relative to $\mathrm{\partial }{I}^n$ in $B$. By [F2], lift that homotopy starting with the first map. On each boundary point the lifted time path lies in the discrete fibre over $b$ and starts at $e$, so it is constant. At time one, the resulting based lift and the second map lift the same cube map and agree at a boundary point; uniqueness in [F1] makes them equal. The lifted homotopy is therefore relative to the boundary, proving injectivity on classes.

Fix $c\in C$, put $y=f(c)$, and let $n\ge 1$. Write $A=f_{\ast }:{\pi }_n(C,c)\to {\pi }_n(X,y)$ and $B=g_{\ast }:{\pi }_n(X,y)\to {\pi }_n(C,g(y))$. The homotopy $gf\simeq {\mathrm{id}}_C$ and [F5] show that $BA=(gf{)}_{\ast }$ is a basepoint-transport isomorphism. Similarly, with $D=f_{\ast }:{\pi }_n(C,g(y))\to {\pi }_n(X,fg(y))$, the homotopy $fg\simeq {\mathrm{id}}_X$ shows that $DB=(fg{)}_{\ast }$ is an isomorphism. Hence $A$ and $B$ are injective. For $v\in {\pi }_n(X,y)$, surjectivity of $BA$ gives $u$ with $BA(u)=B(v)$; injectivity of $B$ then gives $A(u)=v$. Thus $A$ is an isomorphism, even though the original inverse homotopies need not be based.

By [F3], $p_{\ast }$ is a homomorphism; steps 1.1 and 2.1 show it is bijective and hence an isomorphism. Basepoint transport in [F5] makes vanishing independent of the basepoint in a path-connected space. Consequently $E$ is aspherical exactly when $B$ is. For either space that is a CW complex, the $K(G,1)$ criterion follows from [F4] and its specified fundamental-group identification.

For the specified basepoint $x\in X$, choose a path $\eta :y\to x$. By [F5] and step 2.2, ${\beta }_{\eta }^{-1}\circ f_{\ast }:{\pi }_n(C,c)\to {\pi }_n(X,x)$ is an isomorphism for every $n\ge 1$. In degree one, composing with the specified isomorphism to $G$ equips $C$ with the required fundamental-group identification. In degrees at least two, it shows that $C$ is aspherical exactly when $X$ is. Thus if $X$ is aspherical, $C$ is a $K(G,1)$ by [F4]. Conversely, if $X$ is homotopy equivalent to a $K(G,1)$, the same argument with that CW model gives vanishing of the higher groups of $X$. No step identifies the fundamental groups of $E$ and $B$ or asserts finiteness of a CW model.