Show that \(H_c^0(X; G) = 0\) if \(X\) is path-connected and noncompact.
Solution
A degree-zero cocycle with coefficients in \(G\) is a locally constant function \[ f:X\to G. \] If \(X\) is path-connected, every locally constant function is constant. A compactly supported degree-zero class must therefore be represented by a constant function whose support is compact.
If the constant is nonzero, its support is all of \(X\), which is noncompact. Hence the only compactly supported degree-zero cocycle is the zero function. Therefore \[ \boxed{H_c^0(X;G)=0.} \]