Let $X$ be a topological space and let $Y\subset X$ be a subspace. I can write the long exact sequence of relative cohomology $$ H^{k-1}(Y)\xrightarrow{d_*} H^k(X;Y)\xrightarrow{j_*} H^k(X)\xrightarrow{i_*} H^k(Y), $$ where the maps $i_*$ and $j_*$ are those induced in cohomology by the short exact sequence of cochain $$ 0\rightarrow C^k(X;Y)\xrightarrow{j} C^k(X)\xrightarrow{i}C^k(Y)\rightarrow 0,$$ and the map $d_*$ is the usal map in relative cohomology.
Are there conditions equivalent to $j_*$ being injective for every $k$?
In particular, I am working with compact complex manifolds and I consider the spaces of $k$-forms as the complex of cochains.
