I was having some trouble with this concept which makes sense to me intuitively but the understanding of which is not yet fully clear.
Suppose $CS^n$ is the unreduced cone on the n-sphere $S^n$. By unreduced, I mean $CS^n=S^n\times I/S^n\times \{1\}$. We have a map $g:S^n \rightarrow CS^n$ and a map $e: CS^n \rightarrow Z$ for a space $Z$ where $e \circ g$ is null homotopic.
I assumed all maps are based and that $e \circ g$ is homotopic to some constant map $ c:S^n \rightarrow Z$ at the point $e(g(1))$. This gives a triangular diagram (composed of e,g, and c) which commutes up to homotopy. It seems reasonable to conclude that $g$ is null homotopic. I am not fully confident in this.
I was wondering if anyone could shed some light. Any help would be greatly appreciated.
Thanks