# Can this space retract to its boundary?

Consider $$D^2$$, remove two open balls from the interior, and glue two copies of $$S^1$$'s along the boundaries of these two holes, according to the maps $$z \mapsto z^p$$, and $$z \mapsto z^q$$ for distinct primes $$p,q$$. For example, I'm attaching a picture of how it might look for the case $$p=2$$, $$q=3$$:

Can this space be retracted to it's boundary? I have heard this mentioned in passing by someone, but never understood why. Is this even true?

• There is no retraction since the boundary circle is homologically trivial in your space. Jul 25 at 7:46
• What do you mean by the boundary of the space? Jul 25 at 9:28