relative homotopy groups

Question

I study relative homotopy groups and I have a question:
Let $A\subseteq X$ (not necessarily CW complex) and $\pi_{n}(X,A)$. Is it always possible to find a pointed space Y for which $\pi_{n}(X,A)=\pi_{n-1}(Y)$ for all $n\geq 1$ ?

Thank you in advance

Answer 1

It seems that you are thinking of turning the inclusion $i:A \to X$ (of pointed spaces) into a fibration $p:E_i \to X$ and looking at the fibre (or fiber) $F_i$ of $p$ over the base point $x$ of $X$. Then $\pi_n(X,A,x) \cong \pi_{n-1}(F_i,y)$. This idea is important for your study of relative homotopy groups.

There is a wikipedia entry on homotopy fiber though it is slightly inaccurate since it refers to spaces rather than spaces with base point.