I'm brand new to homotopy theory so I'm sure this question is utterly stupid. But anyway I'm trying to understand a proof in the book "Topology and Geometry" by Glen Bredon. This is the proposition:
Let $\partial: \pi_2 (X,A,\ast)\rightarrow \pi_1 (A,\ast)$ be the boundary map from the long exact sequence of homotopy groups. For $\alpha, \beta \in \pi_2 (X,A,\ast)$ we have $(\partial (\alpha))\beta=\alpha \beta \alpha^{-1}$. Where the left hand side means the action of $\pi_1(A,\ast)$ on $\pi_2 (X,A,\ast)$.
My confusion lies in the very first sentence of the proof:
A representative $\mathbb{D}^2\rightarrow X$ of $\beta$ can be taken so that everything maps into the base point except for a small disk and this disk can be placed anywhere. Thus we see that a representative of $\alpha \beta \alpha^{-1}$ can be taken as in the first part of figure VII-7.
But isn't this equivalent to the assumption that the restriction of $\beta$ to $S^1$ is null-homotopic in $\pi_1 (A)$? It seems to me that if $\beta$ lay in the same relative homotopy class as a map which is constant on $S^1$ then any homotopy between these two maps would also provide a contraction of $\beta|_{S^1}$ in $A$.