Let $Z$ be a closed submanifold in some ambient manifold, and $T$ be a tubular neighborhood of $Z$. Assume $dimZ=n$ and $dimT=n+r$. So we have a map $\pi:T\to Z$ with fibers of dimension $r$. Let $[Z]\in H^{r}(T,\mathbb Z)$ be the cohomology class associated to $Z$. Then it claims $$\pi_*[Z]=1$$
I want to know how to prove this?
I can only understand the definitions. The $[Z]$ is defined by the image of $1\in H^0(Z)$ in the composition $$H^0(Z)\xrightarrow[\text{ isomorphism}]{\text{Thom}} H^r(T,T-Z) \to H^r(T)$$ and the $\pi_*$ in $\pi_*[Z]$ should be understood as the Poincare duality of the map $$H_n(T)\to H_n(Z)$$ which is $$H^r(T)\to H^0(Z)$$ But I don't know how to think about $\pi_*[Z]$ then.