Algebraic intersection

Question

Let $D_n$ be $n$-puncture disc and $\pi:\tilde{D_n} \to D_n$ be the infinite cyclic covering. As it is known in this case $H_1(\tilde{D_n};Z)$ is free $(n-1)$-dimensional $Z[t,t^{-1}]$ module. Let $<-,->:H_1(\tilde{D_n},\partial{\tilde{D_n}};Z) \times H_1(\tilde{D_n};Z) \to Z[t,t^{-1}]$ be algebraic intersection operation. I am interesting if it is possible to have $a \in H_1(\tilde{D_n},\partial{\tilde{D_n}};Z)$ such that $<a,b>=0$ for all $b \in H_1(\tilde{D_n};Z)$.