I believe that $S^1\vee S^1$ is the Eilenberg-Mac Lane space $K(\mathbb{Z}\ast\mathbb{Z},1)$. One can prove this by constructing its universal cover and observing that it is contractible.

My question is this:

**Is there a simple homotopy-theoretic proof of this result that does not use connected covers?**

I have never seen one appear in any text. Repeated attemps by myself to use the usual tools (Hilton-Milnor Theorem, examine the fibre of the inclusion $S^1\vee S^1\hookrightarrow S^1\times S^1$, etc...) to prove this elementary result have met with connectivity issues or similar problems.

injectivegroup homomorphisms, then the homotopy colimit of the classifying spaces of those group, $\mathrm{hocolim}_{x \in G} BF(x)$, is a 1-type. This is Theorem 1B.11 in Hatcher's Topology. (The proof builds the universal cover of the colimit.) $\endgroup$ – Omar Antolín-Camarena Jul 11 '16 at 19:29