When the Induced homomorphism on fundamental groups is surjective?

Question

If $X$ is a path-connected topological space and $A$ is a pathconnected subspace of $X$, do we have some "standard" conditions on $X$ or $A$ or the inclusion $i:A\hookrightarrow X$ under which the inclusion $i$ induces an epimorphism $$i_*:\pi_1(A)\longrightarrow \pi_1(X)\; $$ on fundamental groups?

Answer 1

One standard such condition is that $X$ is obtained from $A$ by (iteratively) attaching cells of dimension $\ge 2$. You can find this result in Hatcher's book for example, as a straightforward application of Van Kampen's Theorem.

As a consequence, if $X$ is a CW complex then for any $n \ge 1$ the inclusion of the $n$-skeleton $X^{(n)} \hookrightarrow X$ induces an epimorphism of fundamental groups (for any base point in $X^{(n)}$).

Indeed one can go further: If in addition $X$ is obtained from $A$ by iteratively attaching only cells of dimension $\ge 3$ then the inclusion $A \hookrightarrow X$ induces an isomorphism of fundamental groups. Thus for any $n \ge 2$ the inclusion of the $n$-skeleton of a CW complex induces an isomorphism of fundamental groups.