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 $p=2$, $q=3$: enter image description here

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?

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


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