The rational symmetric signature of manifolds with finite fundamental group (Q1382177)

Let \(M\) be a closed, oriented manifold of dimension \(2k\). The rational intersection form \[ \langle ,\rangle_{\mathbb Q}: H^k(M;\mathbb Q) \times H^k(M;\mathbb Q) \rightarrow \mathbb Q \] is a non-singular \((-1)^{k}\)-symmetric pairing by Poincaré duality. Let \(G\) be a finite group and \(w:G \rightarrow \{\pm 1\}\) a homomorphism. If one assumes in addition that \(M\) has a free \(G\)-action such that \(g_*[M] = w(g)[M]\) for all \(g \in G\), then the intersection form has the invariance property \(\langle g\alpha,g\beta\rangle_{\mathbb Q} = w(g) \langle \alpha,\beta\rangle_{\mathbb Q}\). The main result of the paper shows that the intersection form on \(H^k(M;\mathbb Q)\) has an invariant Lagrangian if either \(w=1\) and \(\text{sign}(M/G)=0\) or \(w \not= 1\) and the Euler characteristic \(\chi(M/G)\) is even. Various results also discuss the realization of these invariants.