algebraic-topologygeneral-topologyklein-bottlemobius-band
Consider the Klein bottle (this can be done by making a quotient space). I want to give a proof of the following statement:
The Klein Bottle is homeomorphic to the union of two copies of a Möbius strip joined by a homeomorphism along their boundaries.
I know what such a Möbius band looks like and how we can obtain this also by a quotient map. I also know how to see the Klein bottle, but I don't understand that the given statement is correct. How do you construct such a homeomorphism explicitly?
Each compact surface (except the sphere) has a standard fundamental polygon representation, and each surface belongs to one of the two categories (non-orientable) $mP^2$ or (orientable) $nT^2$. For example, a torus ($T^2$) has a standard representation $aba^{-1}b^{-1}$, see the following figure:
To construct the connected sum between two surfaces, we cut off a corner from each fundamental polygon, then glue the two polygons along the cut off edge. This is a process of concatenating (or inserting) the standard representation of one surface to another. For example, if you glue two torus handles you get a new representation like:
$$aba^{-1}b^{-1}cdc^{-1}d^{-1}$$
This is the standard representation of a $2T^2$, that is, a genus $2$ torus (a torus with two holes).
Back to your question, a Mobius band is a (real) projective space with a disc cut. It has a triangle representation:
Glue the third edge of this triangle with the torus handle, we get a new representation:
$$abcca^{-1}b^{-1}$$
See the following figure:
This representation has an edge pair $cc$ indicating that it belongs to the non-orientable category, and it has a single vertex class thus it has $6$ edges in its standard representation, implying that it is a $3P^2$. That is we've shown that the connected sum between $T^2$ and $P^2$ is
$$T^2\#P^2=3P^2$$
This crude drawing may be helpful.
Best Answer
Let's directly glue two Möbius strips into a Klein bottle.
Therefore, a Klein bottle is homeomorphic to the glued Möbius strips.