Express the Klein bottle as the cofibre of a map $\kappa$ between (wedges of ) copies of $S^1.$

Question

Express the Klein bottle as the cofibre of a map $\kappa$ between (wedges of ) copies of $S^1.$ Describe the map explicitly.

Could anyone help me in finding this map please?

Answer 1

The Klein bottle $K$ is a non-orientable surface, so by the classification of such is homeomorphic to a connected sum of some number of $\mathbb{R}P^2$. Checking homology we see that $$K\cong\mathbb{R}P^2\#\mathbb{R}P^2.$$ The connected sum is formed by removing a disc from each copy of $\mathbb{R}P^2$ and gluing the two punctured manifolds together along their common boundary $S^1$. Topologically $\mathbb{R}P^2$ is obtained from $\mathbb{R}P^1$ by attaching a $2$-cell along the $2$-sheeted covering $p_2:S^1\rightarrow \mathbb{R}P^1$. We have $\mathbb{R}P^1\cong S^1$ in the standard way, and this identifies the $2$-sheeted covering with the degree $2$ map $\underline 2:S^1\rightarrow S^1$. Hence $\mathbb{R}P^2\cong S^1\cup_{\underline 2} e^2$.

Now we can arrange so that the disc which is to be removed is contained in the interior of the open $2$-cell, and it's easy to see now that $\mathbb{R}P^2\setminus D^2\simeq \mathbb{R}P^2\setminus\{p\}$ deformation retracts onto the $1$-skeleton $\mathbb{R}P^1\cong S^1$. Thus the connected sum $\mathbb{R}P^2\#\mathbb{R}P^2$ may be described topologically as the pushout $\require{AMScd}$ \begin{CD} S^{n-1}@> -\underline 2 >> S^1\\ @V\underline 2V V @VV V\\ S^1 @>>> \mathbb{R}P^2\# \mathbb{R}P^2. \end{CD} Here I'm forming the $-\underline 2$ using the suspension structure on $S^1$. In this case, for identifying the homotopy type of $\mathbb{R}P^2\# \mathbb{R}P^2$ it doesn't matter wether you insert this sign change or not, but it will make what I write next more correct and easier. In any case, the sign does appear if you work with oriented manifolds and ask for an oriented connected sum (reason: when you glue the boundaries of the two removed discs together, you have to reverse the orientation on one).

Now, with the above pushout we take the two maps $\underline 2$ and $-\underline 2$, flip the sign on one, and then add them together using the suspension structure to get the map $$\varphi=i_1\underline 2+i_2\underline 2:S^1\rightarrow S^1\vee S^1$$ where $i_j:S^1\hookrightarrow S^1\vee S^1$ is the inclusion into the $j^{th}$ wedge summand. Then, being careful to apply a standard result, we see that the cofibre of $\varphi$ is homotopy equivalent to $\mathbb{R}P^2\#\mathbb{R}P^2\cong K$. i.e. there is a (homotopy) cofibre sequence $$S^1\xrightarrow{\varphi}S^1\vee S^1\rightarrow K.$$ You can check using homology that we do in fact get the Klein bottle from this.