Let $ M $ be a smooth and compact manifold with boundary $ \partial M = X \times F $ on which the structure of a smooth locally trivial bundle $$ \pi: \partial M \longrightarrow X $$
where $ X $ - the base  $F$- the fiber are smooth compact manifolds without boundary. By sending $\partial M$ to $X$ we obtained  a new manifold called $N$. This manifold can be not smooth. How to define a map $$I : H^{n-k}_{dR}(M)\longrightarrow H_{k}(N)$$  when N is not smooth??