Plane models from the "word"

Question

I have a "word" for a plane model $abacdc^{-1}db^{-1}$. From what I reckon, it's a torus. But I am not too sure of it. I sketched it up and did some "adjustments". Could it be a projective plane perhaps?

Answer 1

Since your question lacks some details on what you know and what you have tried, I will make some assumptions about what you know in order to give you the quickest route to an answer, which is to apply the theorem on the classification of surfaces.

To apply that theorem, you need to compute the orientability and the Euler characteristic of the surface.

Draw an octagon and label its sides using the word, with a backwards arrow on a side labelled with a $-1$ exponent and a forwards arrow otherwise.

To determine orientability, since the two $a$ sides of the octagon are oriented in the same direction, the surface is nonorientable. That immediately rules out your reckoning of a torus.

To determine the Euler characteristic, first compute the vertex cycles and then count how many cycles there are. In this case, all 8 vertices are in a single vertex cycle, as you can verify easily. So the surface you get by gluing up the octagon has $1$ vertex from that single vertex cycle, $4$ edges from pairing up the $8$ sides of the octagon, and $1$ face from the octagon itself.

Thus the Euler characteristic is $1-4+1=-2$. By the theorem on the classification of surfaces there is just one nonorientable surface of Euler characteristic $-2$, and it is homeomorphic to a connected sum of $4$ projective planes.