Q:

How many Vertices, Edges, Faces are there in these Diagrams?

Apologies for the really basic question, however, I don't really understand how to count edges/faces for these types of diagrams with vertices "identified".
E.g. For Figure 3.11, Clearly $V=4$, but $E$ and $F$ I have no idea how to count. In particular, there are two edges "bc", should we count both or just one?
Also, there are two faces abc.
This is supposedly a Mobius Strip, so $\chi=V-E+F=0$.
Thanks!

A:

You write '…with vertices "identified"…'. 
But there is missing information: are edges along the outer boundary also supposed to be identified? For example, in Figure 3.11, are we supposed to identify the $a-b$ edge on the left side of the figure and the $a-b$ edge on the right side of the figure to a single edge? Without more context, it's impossible to tell. Maybe someone just wrote some letters on Figure 3.11 and no identifications are intended whatsoever.
Now, generally speaking that there is a convention in these kinds of problems: a pair of outer boundary edges with identical endpoint labels is supposed to be identified to a single edge. Also, no face identifications are intended. You can find this convention spelled out in various textbook sources, and my guess is that this is the intended convention in the book that you took these pictures from, although maybe you have to read between the lines to be sure that this is the intended convention. 
By the way, if you follow these conventions you may or may not obtain a "simplicial complex", but you will always obtain a "$\Delta$-complex" (I'm using terminology from Hatcher's book on algebraic topology).
If we follow this convention then to count edges after identifications goes like this. 
In Figure 3.11, the $a-b$ edges on the left and right are identified to 1 edge, and those are the only identifications. Thus, out of the nine visible edges in Figure 3.11, two of those nine are identified to 1 edge, the remaining seven of those nine make 7 edges after identifications, for a total of $1+7=8$ edges after the identifications have been carried out. The number of faces after identification equals $4$.
Figure 3.12 is very similar, with a total of $8$ edges and $4$ faces after identifications.
In Figure 3.13, the left and right two $a-d$ edges are identified to 1 edge, the left and right $d-c$ edges are identified to $1$ edge, the top and bottom $c-b$ edges are identified to $1$ edge, and the top and bottom $b-a$ edges are identified to $1$ edge. None of the remaining eight interior edges are identified to anything, making an additional 8 edges after identifications. This gives a total of $1+1+1+1+8=12$ edges after identification. The number of faces after idenfication equals $8$. 
I'll leave Figure 3.14 for you.

