I would like to know what the following spaces $X$ and $Y$ look like. More precisely, I want to know if they are homeomorphic to some other known spaces. I define $X$ and $Y$ as a quotient of the square $[0,1]^2$ by gluing the edges (red with red, blue with blue):
$\qquad\qquad\qquad\quad$
In other words, $X=[0,1]^2 / \sim$ and $Y=[0,1]^2 / \sim'$ where $\sim$ is the equivalence relation defined by
$$(x,s) \sim (y,t) \iff (x,s)=(y,t) \;\text{ or }\;
s=0, y=1, t = 1-x \;\text{ or }\;
s=1, y=0, x = 1-t $$
and $\sim'$ is the equivalence relation defined by
$$(x,s) \sim' (y,t) \iff (x,s)=(y,t) \;\text{ or }\;
s=0, y=1, t = x \;\text{ or }\;
s=1, y=0, x = 1-t $$
I tried to see with a handkerchief what I could obtain when gluing the edges as shown on the diagrams, but it was quite unsuccessful. I believe that I would obtain non-orientable spaces, maybe a wedge of projective space and something else. I failed to understand what $X$ and $Y$ are homeomorphic to.
However, I know what the following similar diagrams give :
$\qquad\quad$
$\qquad\qquad\qquad\qquad$
Any suggestion will be appreciated, thanks in advance!
Best Answer
To see that the first space is homeomorphic to $\Bbb R \mathrm P^2$, cut it along the top-left to bottom-right diagonal and glue together the blue edges. You will end up with the standard representation of $\Bbb R\mathrm P^2$ in terms of a square with face identifications.
Your second space is a disk (2-cell) glued to a bouquet of two circles $a,b$ (after identifying the loops pairwise) along the loop $a^2b^2$. This is the Klein bottle.