TITLE: identify the topological type obtained by gluing sides of the hexagon
QUESTION [5 upvotes]: Identify the topological type obtained by gluing sides of the hexagon as shown in the picture below


Clearly the boundary is encoded by the word $abcb^{-1}a^{-1}c$
I do not understand how the surface is glued together - could you help me, please?
Edit: This answer is a Klein bottle (see answer below)
Similar problem: see Is this octogon topologically equivalent to the Klein Bottle?

REPLY [3 votes]: You should get the Klein bottle:  

By combining arrows $a$ and $b$ into a single red arrow, and arrows $a^{-1}$ and $b^{-1}$ into another red arrow, we get the first figure below. Now we must glue the red sides together, and blue sides together, so that the arrows line up.