# Relative Homotopy Groups for Covering Spaces

Given a covering map $p\colon\tilde X\to X$, I know that $\pi_n(\tilde X,\tilde x)\cong\pi_n(X,x)$ for $n>1$. I want to generalize this to relative homotopy groups, i.e. given $A\subseteq X$ and $\tilde A=p^{-1}(A)$, I would like to show that $\pi_n(\tilde X,\tilde A,\tilde x)\cong\pi_n(X,A,x)$ for $n>1$. By using the long exact sequence for relative homotopy groups and the five lemma, I can show that the result holds for $n>2$, but the $n=2$ case eludes me. Explicitly, I have two exact sequences:

$\cdots\to\pi_n(A,x)\to\pi_n(X,x)\to\pi_n(X,A,x)\to\pi_{n-1}(A,x)\to\pi_{n-1}(X,x)\to\cdots$

and

$\cdots\to\pi_n(\tilde A,\tilde x)\to\pi_n(\tilde X,\tilde x)\to\pi_n(\tilde X,\tilde A,\tilde x)\to\pi_{n-1}(\tilde A,\tilde x)\to\pi_{n-1}(\tilde X,\tilde x)\to\cdots$

If $n>2$, I can use the five lemma and what I already know about higher homotopy groups for covering spaces to conclude the result. This argument fails for $n=2$ however. Is a separate argument needed for this case? Is the statement even true when $n=2$?

## 1 Answer

I don't think you should use exact sequences in this case even though they're always very tempting. Better than proving only that the spaces are isomorphic in the abstract, you should use the homotopy lifting property for covering spaces, on maps from $S^n$ to $X$, to prove that the covering map induces an isomorphism on higher homotopy groups. To do this use the fact that $S^n$ is simply connected for $n>1$; Wikipedia summarizes the argument here.