Let $M^{4k-1}$ be an oriented differentiable or $PL$ manifold, $t$ a stable homotopy-parallelization of $M$ and $e$ the linking bilinear form on $T_{2k-1}(M)$ torsion subgroup of $H_{2k-1}(M)$.  One constructs a quadratic form $q^t : T_{2k-1} (M) \to \mathbb{Q}/\mathbb{Z}$ with associated bilinear form $e$.

Let $A(M,t)\in \mathbb{Z}/8$ be the Arf invariant of $q^t$ ; one studies how $A(M,t)$ depends on $t$ and one has the formula for the signature

$I(N)\equiv < L(\tau_{t} ) , [N] > - A(M,t)$ mod.8

where $N$ is a manifold of dimension $4k \ge 12$ with boundary $M$ , $t$ a $SPL$ trivialization of $\tau_M$ and $L(\tau_{t})$ the Hirzebruch class of the bundle on $N/M$ defined from $\tau_N$ by $t$.