2 problems

Let $E$ be an acyclic flat complex vector bundle over a closed manifold $M$, so that $H^\bullet(M,E)=0$, and suppose that $E$ admits a non-degenerate complex-valued symmetric bilin…

Let $X$ be a closed $3$-manifold with $b_1(X)>0$, let $o$ be a homology orientation, and let $\operatorname{Eul}(X)$ be the set of Euler structures. Turaev’s torsion defines a map…