I am reading these introductory notes to cobordism theory, however cobordism theory is not required to understand my question. I just need help understanding a certain map.
On page number 8 the author claims that there is a map $pr_1:MO(k) \wedge X_+ \rightarrow MO(k)$ where $MO(k)$ is just some based topological space and $X_+$ is $X \sqcup *$. I understand the existance of this map.
However, they also claim on the top of page 9 that there exists a map $pr_2:MO(k) \wedge X_+ \rightarrow X$, this confuses me since $MO(k) \wedge X_+ = (MO(k) \times X)/(\{ * \} \times X)$ thus $pr_2$ should correspond to a map $pr_2:MO(k) \times X \rightarrow X$ which maps the subspace $\{*\} \times X$ to a single point in $X$ and the obvious choice of $pr_2$, namely $(m,x) \mapsto x$, obviously does not.
This doesn't make sense to me. Am I just missing something? It seems unlikely that $pr_2$ is a typo on behalf of the author since it makes perfect sense to wish for such a map to exist given the context.
Thanks in advance!