I'm working through an introductory homotopy theory book (Arkowitz) for self-study, and I'm a bit stuck on the following exercise. Not a lot of machinery is available at this point in the book. I'm not seeing how to leverage the $(n-1)$-connectivity or the dimensional restrictions. Perhaps I'm missing something obvious:
Let $X$ be an an $(n-1)$-connected space with $n \geq 2$. Prove that if $dim\ X \leq 2n-1$, then there is a comultiplication on $X$. If $dim\ X \leq 2n-2$, prove that any two comultiplications on $X$ are homotopic.
For context, I am including an image of the previous exercise (2.6) and a possibly relevant proposition (2.4.6, also referred to as 8.2.1) from the chapter containing the exercise.
Just to close this out, I'll informally sketch an answer based on the comments I received. We regard $X \vee X$ as the sub-complex of $X \times X$ with support $X \times \{*\}\space \cup \{*\} \times X$, where $i: X\vee X\longrightarrow X \times X$ is the inclusion map. Since $X$ is $(n-1)$-connected we may assume, without loss of generality (Cor $2.4.10$), that $X^{(n-1)}$ (the $(n-1)$ skeleton of $X$) $=\{*\}$. Then $(X \vee X)^{(2n-1)} = (X\times X)^{(2n-1)}$. Thus $i$ is a $2n-1$ equivalence.
By proposition $2.4.6$, $i_*: [X, X\vee X]\longrightarrow [X, X\times X]$ is surjective when $dim\space X \leq 2n-1$ and is a bijection when $dim\space X \leq 2n-2$.
Let $\Delta:X\longrightarrow X\times X$ be the diagonal map. When $i_*$ is surjective, $\exists c\in [X, X\vee X]$ with $i \cdot c$ homotopic to $\Delta$. Then by exercise $2.6$, $c$ is a comultiplication.
When $i_*$ is a bijection, such a $c$ is unique (up to homotopy).