  Let $B$ be a Banach $A-bimodule$. We introduce the weak topological centers
of left module action and we show it by $\tilde{{Z}}^\ell_{B^{**}}(A^{**})$.
For a compact group, we show that
$L^1(G)=\tilde{Z}_{M(G)^{**}}^\ell(L^1(G)^{**})$ and on the other hand we have
$\tilde{Z}_1^\ell{(c_0^{**})}\neq c_0^{**}$. Thus the weak topological centers
are different with topological centers of left or right module actions. In this
manuscript, we investigate the relationships between two concepts with some
conclusions in Banach algebras. We also have some application of this new
concept and topological centers of module actions in the cohomological
properties of Banach algebras, spacial, in the weak amenability and $n$-weak
amenability of Banach algebras.
