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 is given $$ \pi: \partial M \longrightarrow X $$ where the base $ X $ and the fiber $F$ are smooth compact manifolds without boundary. Consider the equivalence relation on the set $M$ \begin{equation} z\sim z^{\prime} \Longleftrightarrow z = z^{\prime} \quad \text{or} \quad (z, z^{\prime} \in \partial M \quad \text{and} \quad \pi(z) = \pi(z^{\prime})).\end{equation} We define the topological space $ N = M / \sim $ as the quotient space of the manifold M with respect to the equivalence above. So the manifold $N$ can be not smooth sometimes. How to define the map $I : H^{n-k}_{dR}(M)\longrightarrow H_{k}(N)$ when N is not smooth?