@BalarkaSen when you say the boundary map is literally the boundary, is that just for manifolds or is there a more general statement.
I'm can only think of the following theorem I'm given: Say $M$ is smooth compact orientable $n$-dimensional, so it's boundaries are closed smooth compact orientable n-1-dimensional. Then under the boundary map in the LES, the fundamental class of $H_n(M, \partial M)$ gets mapped to the sum of the fundamental classes of the boundary components
(or, equivalently, gets mapped to a vector with each entry one fundamental class)