on natural homomorphisms of local cohomology modules

‎let $m$ be a non-zero finitely generated module over a commutative noetherian local ring $(r,mathfrak{m})$ with $dim_r(m)=t$‎. ‎let $i$ be an ideal of $r$ with $grade(i,m)=c$‎. ‎in this article we will investigate several natural homomorphisms of local cohomology modules‎. ‎the main purpose of this article is to investigate when the natural homomorphisms $gamma‎: ‎tor^{r}_c(k,h^c_i(m))to kotimes_r m$ and $eta‎: ‎ext^{d}_r(k,h^c_i(m))to ext^{t}_r(k‎, ‎m)$ are non-zero where $d:=t-c$‎. ‎in fact for a cohen-macaulay module $m$ we will show that the homomorphism $eta$ is injective (resp‎. ‎surjective) if and only if the homomorphism $h^{d}_{mathfrak{m}}(h^c_{i}(m))to h^t_{mathfrak{m}}(m)$ is injective (resp‎. ‎surjective) under the additional assumption of vanishing of ext modules‎. ‎the similar results are obtained for the homomorphism $gamma$‎. ‎moreover we will construct the natural homomorphism $tor^{r}_c(k‎, ‎h^c_i(m))to tor^{r}_c(k‎, ‎h^c_j(m))$ for the ideals $jsubseteq i$ with $c = grade(i,m)= grade(j,m)$‎. ‎there are several sufficient conditions on $i$ and $j$ to provide this homomorphism is an isomorphism.