If \(X\) is a finite connected graph with \(\pi_1(X)\) free on \(g > 0\) generators, show that \(H^n(X; \mathbb{Z}[\pi_1 X])\) is zero unless \(n = 1\), when it is \(\mathbb{Z}\) when \(g = 1\) and the direct sum of a countably infinite number of \(\mathbb{Z}\)’s when \(g > 1\). [Use Proposition 3H.5 and compute \(H_c^n(\tilde{X})\) as \(\varinjlim H^n(\tilde{X}, \tilde{X} - T_i)\) for a suitable sequence of finite subtrees \(T_1 \subset T_2 \subset \cdots\) of \(\tilde{X}\) with \(\bigcup_i T_i = \tilde{X}\).]
Solution
Let \[ \pi=\pi_1(X), \qquad p:\widetilde X\to X \] be the universal cover. Proposition 3H.5 identifies \[ H^n(X;\mathbb Z[\pi])\cong H_c^n(\widetilde X;\mathbb Z). \] Since \(X\) is a finite connected graph with free fundamental group of rank \(g>0\), \(\widetilde X\) is a locally finite tree.
Choose an exhaustion by finite connected subtrees \[ T_1\subset T_2\subset\cdots, \qquad \bigcup_iT_i=\widetilde X. \] Compactly supported cohomology is \[ H_c^n(\widetilde X) \cong\varinjlim_i H^n(\widetilde X,\widetilde X-T_i). \] Because \(\widetilde X\) is contractible and one-dimensional, the long exact sequence of the pair gives \[ H^n(\widetilde X,\widetilde X-T_i)=0\qquad(n\ne1), \] while \[ H^1(\widetilde X,\widetilde X-T_i) \cong\widetilde H^0(\widetilde X-T_i). \] If \(c_i\) is the number of components of \(\widetilde X-T_i\), this is free abelian of rank \(c_i-1\).
If \(g=1\), the universal cover is a line. For large interval subtrees \(T_i\), the complement has exactly two components, and the transition maps identify the corresponding generators. Hence \[ H_c^1(\widetilde X)\cong\mathbb Z. \]
If \(g>1\), the universal covering tree has branching. Choose the exhaustion so that each \(T_{i+1}\) is obtained by adding a finite layer of adjacent edges. Then the maps \[ \widetilde H^0(\widetilde X-T_i)\to \widetilde H^0(\widetilde X-T_{i+1}) \] are split injections: each old complementary component breaks into finitely many new components, and choosing one distinguished descendant in each component gives a splitting. The ranks \(c_i-1\) tend to infinity because the tree has infinitely many ends and branching repeats indefinitely. Thus the direct limit is a free abelian group on a countably infinite set of generators.
Therefore \[ \boxed{H^n(X;\mathbb Z[\pi])=0\quad(n\ne1),} \] and \[ \boxed{H^1(X;\mathbb Z[\pi])\cong \begin{cases} \mathbb Z,&g=1,\\ \bigoplus_{\mathbb N}\mathbb Z,&g>1. \end{cases}} \]