Explicit construction of Eilenberg-Maclane spaces with n=1

Question

Is there any examples of explicit construction of Eilenberg-Maclane spaces $K(G,1)$ for concrete groups except for G=$\mathbb Z$ and $\mathbb Z_n$? I know about general simplicial bar construction, but is there anything more concrete except for sphere and lens spaces?

Answer 1

I'm not sure what exactly you're looking for; $K(G, 1)$ is not a nice manifold or even a finite complex in general. (For example, if $G$ contains torsion, then $K(G, 1)$ can't be homotopy-equivalent to a finite-dimensional complex for cohomological reasons.) For some other explicit examples, though:

• A closed hyperbolic manifold $X$ has a contractible universal cover $\tilde X$ by the Cartan-Hadamard theorem, so $X = K(\pi_1 X, 1)$.
• Similarly, the complement of a knot $K$ in $S^3$ is a $K(G, 1)$.
• The space $K(F_n, 1)$, where $F_n$ is the free group of rank $n$, is the wedge sum of $n$ copies of $S^1$.
• In higher dimensions, $K(\mathbb{Z}, 2) = \mathbb{CP}^\infty$.
• By the Dold-Thom theorem, the infinite symmetric product of $S^n$ is a $K(\mathbb{Z}, n)$.
• Similarly, it's possible to write down a more specific description of $K(G, 1)$ for $G$ a braid group or pure braid group.

Answer 2

One way of thinking about this is to say it is about constructing resolutions of the group $G$ from information about $G$, for example a presentation of $G$. You could look here at the work of Graham Ellis on Homological Algebra Programming.

I think the question is related to the spherical space form problem on periodic resolutions and groups which act freely on spheres, since that will give some explicit examples.