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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 the $ 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 relation above. Informally speaking, $ N $ is obtained from $ M $ (by contracting the fibers of the bundle $ \pi $ to points). The set $ N $ is a disjoint union $ N = X \sqcup M^{\circ} $ of the manifold $ X $ and the interior $ M^{\circ} $ of $ M $. The natural projection of $$ p: M \longrightarrow N $$ coincides with the identity map on $ M ^ {\circ} $ and the projection $ \pi $ on $ \partial M $. 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 $F$ is not a singleton?

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 the $ 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 relation above. Informally speaking, $ N $ is obtained from $ M $ (by contracting the fibers of the bundle $ \pi $ to points). The set $ N $ is a disjoint union $ N = X \sqcup M^{\circ} $ of the manifold $ X $ and the interior $ M^{\circ} $ of $ M $. The natural projection of $$ p: M \longrightarrow N $$ coincides with the identity map on $ M ^ {\circ} $ and the projection $ \pi $ on $ \partial M $. So the manifold $N$ can be not smooth sometimes. The pair $(M,\pi)$ is called a manifold with fibered boundared. How to define the map $I : H^{n-k}_{dR}(M,\pi)\longrightarrow H_{k}(N)$ when $F$ is not a singleton?

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?

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 the $ 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 relation above. Informally speaking, $ N $ is obtained from $ M $ (by contracting the fibers of the bundle $ \pi $ to points). The set $ N $ is a disjoint union $ N = X \sqcup M^{\circ} $ of the manifold $ X $ and the interior $ M^{\circ} $ of $ M $. The natural projection of $$ p: M \longrightarrow N $$ coincides with the identity map on $ M ^ {\circ} $ and the projection $ \pi $ on $ \partial M $. 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 $F$ is not a singleton?

Let $ M $ be a smooth and compact manifold with boundary $ \partial M = X \times F $ and the projection $$ \pi: \partial M \longrightarrow X $$ 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?

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?