  We show that an exact Lagrangian cobordism $L\subset \mathbb R \times P
\times \mathbb R$ from a Legendrian submanifold $\Lambda\subset P\times \mathbb
R$ to itself satisfies $H_i(L;\mathbb F)=H_i(\Lambda;\mathbb F)$ for any field
$\mathbb F$ in the case when $\Lambda$ admits a spin exact Lagrangian filling
and the concatenation of any spin exact Lagrangian filling of $\Lambda$ and $L$
is also spin. The main tool used is Seidel's isomorphism in wrapped Floer
homology. In contrast to that, for loose Legendrian submanifolds of
$\mathbb{C}^n \times \mathbb R$, we construct examples of such cobordisms whose
homology groups have arbitrary high ranks. In addition, we prove that the front
$S^m$-spinning construction preserves looseness, which implies certain
forgetfulness properties of it.
