Is there any 'general' topological invariant to tell the difference between $M$ and $N$, where $M$ has homotopy type of a closed manifold and $N$ has homotopy type of a manifold with boundary. 


I meant something like homotopy group/ homology group can 'detect' the obstruction of being homotopic to a closed manifold.



Topological description of Manifold with boundary
If $M$ is a closed manifold, then you can always cross it with an interval to get a
manifold with non-empty boundary that is homotopic to $M$. So the distinction between the two possibilities (closed or non-closed) is not so well-defined in the homotopy category.


On the other hand, suppose that we fix the dimension $n$ of the manifolds we want to consider.
Then,
the most basic difference between closed manifolds and those with boundary is the top dimensional homology, namely: a closed orientable connected manifold of dimension $n$ has $H\_n(M) = {\mathbb Z}$,
while on the other hand, if $N$ has non-empty boundary, then $H\_n(N) = 0$. 


