Connected sum of four surfaces to equal 2 klein bottles

Question

Topology: Write the surface 2K (K=Kleinbottle) as the connected sum of four (not necessarily distinct) surfaces. In how many different ways can you do this?

I know that 1 klein bottle= 2 projective planes then we can write,

p#p#p#p and this will be one way. Could there be any other way to connect a sum of four surfaces to equal 2 klein bottles?

Answer 1

If you require it to be 4 surfaces, where none of them are spheres, then there's only one way.

But if you allow spheres, then you can use $P\#P\#P = T^2 \# P = K \# P$.

Answer 2

There are a lot of other ways. For instance, you could take three spheres and one $K\#K$. To systematically find all the ways, you can use the fact that a closed surface is determined up to homeomorphism by its Euler characteristic and whether it is orientable. Moreover, if $S$ and $T$ are closed surfaces then $\chi(S\#T)=\chi(S)+\chi(T)-2$ and $S\#T$ is orientable iff both $S$ and $T$ are orientable. Think about what restrictions this gives you if you want to have four surfaces whose connected sum is nonorientable and has Euler characteristic $\chi(K\# K)=-2$.