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Let $M$ be a topological manifold with boundary $\partial M$. Let $N$ be a quotient of the topological space $\{0,1\}\times M$, two copies of $M$, with respect to the relation $(x,0)\sim (x,1)$ for all $x\in\partial M$. Then $N$ is a topological manifold without boundary.

Similar question I found is this one: Poincare duality

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Hint: think about the plane as the union of the (closed) upper half-plane $U$ and the (closed) lower half-plane $L$ (which is a homeomorphic copy of $U$). Each of those half-planes is a manifold with boundary $\partial U = \partial L$, the $x$-axis. Given $p$ on the $x$-axis, and charts around $p$ in $U$ and $L$, you can glue pieces of those charts together to get a chart around $p$ in $U \cup L$and so make $U \cup L$ into a manifold without boundary (when you include all the charts around points in the interiors of $U$ and $L$). Now generalise with $M$ and $\partial M$ in place of $U$ and the $x$-axis.

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