Isomorphism relative homotopy groups

Question

Suppose that we have $Y \subset X$ topological space such that $$ \pi_i(X,Y)=0$$ for all $0 \le i < k$. How can I prove that the homomorphism induced by inclusion $i: Y \hookrightarrow X$, say $i^*: \pi_i(Y) \to \pi_i(X)$ is an isomorphism for all $0 \le i \le k-2$?

Answer 1

To build on Stefan H's comment. As in Hatcher, if one defines $\pi_n(X,x_0)$ to be the set of homotopy classes of maps $f:(I^n,\delta I^n)\rightarrow (X,x_0)$, we can extend this definition to $\pi_0$ by taking $I^0$ to be a point and $\delta I^0$ to be empty.

This actually agrees with the homotopy classes of pointed maps $[S^0,X]$ that you stated ArthurStuart. $\pi_0$ then becomes the set of path-components of $X$, notice that it doesn't have a group structure, however the sequence that Stefan posted is still exact until $\pi_0(X,x_0)$.

As Stefan said, it now depends how you define $\pi_0(X,A,x_0)$, but since you are given that all $(\lt k)$ the relative homotopy groups of your spaces disappear, then this still gives you the isomorphism $i^∗:\pi_0(Y)→\pi_0(X)$ .